A THESIS. Submitted by MAHALINGA V. MANDI. for the award of the degree of DOCTOR OF PHILOSOPHY

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1 LINEAR COMPLEXITY AND CROSS CORRELATION PROPERTIES OF RANDOM BINARY SEQUENCES DERIVED FROM DISCRETE CHAOTIC SEQUENCES AND THEIR APPLICATION IN MULTIPLE ACCESS COMMUNICATION A THESIS Submitted by MAHALINGA V. MANDI for the award of the degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING Dr. M.G.R. EDUCATIONAL AND RESEARCH INSTITUTE UNIVERSITY (Declared u/s 3 of the UGC Act, 1956) CHENNAI JANUARY 2013

2 LINEAR COMPLEXITY AND CROSS CORRELATION PROPERTIES OF RANDOM BINARY SEQUENCES DERIVED FROM DISCRETE CHAOTIC SEQUENCES AND THEIR APPLICATION IN MULTIPLE ACCESS COMMUNICATION A THESIS Submitted by MAHALINGA V. MANDI for the award of the degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING Dr. M.G.R. EDUCATIONAL AND RESEARCH INSTITUTE UNIVERSITY (Declared u/s 3 of the UGC Act, 1956) CHENNAI JANUARY 2013

3 ii BONAFIDE CERTIFICATE Certified that this thesis titled Linear Complexity and Cross Correlation Properties of random binary sequences derived from discrete chaotic sequences and their application in multiple access communication is the bonafide work of Mr. Mahalinga V. Mandi who carried out the research under our supervision. Certified further, that to the best of our knowledge the work reported herein does not form part of any other thesis or dissertation on the basis of which a degree or award was conferred on an earlier occasion on this or any other candidate. Dr. R. Murali Co-Supervisor Professor, Department of Mathematics, Dr. Ambedkar Institute of Technology, Bangalore Dr. K. N. Hari Bhat Supervisor Prof. & Head, Dept. of ECE, Nagarjuna College of Engineering & Technology, Bangalore

4 iii DECLARATION This is to certify that the thesis titled Linear Complexity and Cross Correlation Properties of random binary sequences derived from discrete chaotic sequences and their application in multiple access communication submitted by me to the Dr. M.G.R. Educational and Research Institute University for the award of the degree of Ph.D is a Bonafide record of research work carried out by me under the supervision of Dr. K. N. Hari Bhat and Dr. R. Murali. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma. Signature of the Research Scholar (MAHALINGA V. MANDI)

5 iv Dedicated to the Beacon of light of my life and Fountain of Knowledge Dr. K. N. HARI BHAT who guided me in right path in completing this Epic Research Fruitfully

6 v ABSTRACT Set of binary sequences with good pair wise cross-correlation finds application in code division multiple access (CDMA) system. Normally in these schemes, since crosscorrelation has a minimum value but not zero, if the number of simultaneous users increases at a time, there is degradation in the quality of the recovered signal. Chaotic sequences are highly uncorrelated even when their initial values differ by a small value and hence can be used as address sequences in spread spectrum communication. In chaotic spread spectrum systems, different user may be assigned different sequences generated with different initial conditions. Chaotic functions exhibit cryptographically desirable qualities for secured data communication also. From implementation point of view binary sequences are desirable. This investigation deals with generation of chaotic sequences and deriving discrete sequences over Z 4 and binary sequences derived from sequences over Z 4, using binary conversion and three polynomial mappings. Different chaotic maps like Logistic map, Tent map, Cubic map, Bernoulli map and Quadratic map are considered in this work. Chaotic sequences over finite fields of the form GF(2 m ) are defined and binary sequences are derived from them and their properties are investigated. Study of Linear Complexity (LC) and Cross Correlation (CCR) properties of all the derived binary sequences and their suitability in CDMA are carried out. There are (2 L + 1) Gold sequences of length (2 L -1) bits that can be generated using two L stage Linear Feedback Shift Register (LFSR). The maximum Linear Complexity of these sequences is 2L. Similarly large set Kasami sequences have length (2 L -1) for even L and have maximum linear complexity 5L/2. Number of large set Kasami sequences is 2 L/2 (2 L +1). Small set Kasami sequences has linear complexity 3L/2 having length (2 L -1) and the number of sequences in the set is 2 L/2.

7 vi A scheme is proposed by which it is possible to derive binary sequences using chaotic sequences whose CCR and LC properties are better than Gold and Kasami sequences. It is shown that some of the segments of generated sequences using the proposed method have linear complexity greater than that of Gold and small set Kasami sequences of same length. It is also shown that there are segments of sequences whose pairwise maximum magnitude of CCR value less than that of Gold or large set Kasami sequences. The number of such sequences is less than the number of Gold sequences or large set Kasami sequences of same length. Further it is shown that the number of segments of sequences of length (2 L -1) having peak magnitude of pairwise CCR value same as that of Gold sequences is greater than the number of Gold sequences. A smaller set of Segments of binary sequences derived from chaotic sequences over Z 4 using four mappings and binary sequences derived from chaotic sequence over GF(2 m ) having peak magnitude of pairwise cross correlation value denoted by, much lower than CCR values of Gold and Kasami sequences, satisfies the Welch bound and Sidelnikov bound better than Gold and Kasami sequences of same length. The BER performance of CDMA systems for a given depends on the number of simultaneous users. This is because, due to non-zero pairwise CCR values of code sequences, the multiple access interference (MAI) increases as number of simultaneous users increases. Hence using the proposed binary sequences with low pairwise CCR value as code sequences, for a given transmitter power and band width, the BER performance is better compared to using Gold or large set Kasami sequences as code sequences. On the other hand for a given BER performance and band width, transmitter power can be reduced using proposed sequences as code sequences. It is shown that similar properties are exhibited using binary sequences derived from the chaotic sequences over finite field GF(2 m ). Key Words: Chaotic Functions, Chaotic Sequences over Z 4, Chaotic Sequence over Finite Field GF(2 m ), Binary Conversion, Polynomial Mapping, Cross Correlation, Linear Complexity, Code Division Multiple Access and Bit Error Rate performance.

8 vii ACKNOWLEDGEMENTS I am grateful to Dr. M. G. R. Educational and Research Institute University, Chennai and Dr. Ambedkar Institute of Technology, Bangalore, for giving me an opportunity to work on this thesis. I would like to express my heartfelt thanks to Thiru A. C. Shanmugam, Founder Chancellor, Er. A. C. S. Arunkumar, President, Prof. Dr. K. Meer Mustafa Hussain, Vice Chancellor and Dr. A. Thirunavukarasu, Dean Research of Dr. M. G. R. Educational and Research Institute University, Chennai for their guidance and encouragement at all stages of my work. It's been my great fortune to work in the conducive atmosphere which is formulated by my guide & mentor Dr. K. N. Hari Bhat, Dean, Professor and Head, Department of E&C (PG), Nagarjuna College of Engineering and Technology, Bangalore. This dissertation wouldn't have been possible without his. His extended valuable support in the completion of my thesis is inevitable. He has been my inspiration to transit my enthusiasm, creativity & experience into a position where I could provide the strategic & tactical explanation for the thesis submitted. I want to thank my Guide for all the suggestion & motivation provided to me. It has been an honorary privilege for me to work with his support & assistance. I owe my foremost gratitude for him throughout my life for the blessings he has showered on me.

9 viii I am deeply indebted to my research co-supervisor Dr. R. Murali, Professor, Department of Mathematics, Dr. Ambedkar Institute of Technology, Bangalore who has provided me with proper advice, insightful and thought provoking feedbacks in the course of this study, though sometimes with gentle nudge, to propel me to complete my thesis on time. I thank him for his forbearance in supervising me. I would like to thank Dr. S. Ravi, Professor and Head, Department of E& C, Dr. M.G.R. Educational and Research Institute University, for his advice and encouragement. I would also like to thank the Administrative staff for assisting me in complicated administrative matters, especially Prof. R. Amutha and L. Sowdammal. I am grateful to the Management and Principal of Dr. Ambedkar Institute of Technology, Bangalore, having given kind co-operation in completing my Research program. I am grateful to all my colleagues and friends in the department for advice, assistance, and encouragement especially to Sri Shivaputra and Sri Chetan S. I would like to express my heartfelt thanks to the Management, Principal and Staff of Nagarjuna College of Engineering and Technology, Bangalore, for having rendered valuable guidance and encouragement for my Research work. My special thanks to Sri Dr. Ramachandra, Professor and Head, Department of Chemistry, Reva Institute of Technology and Management, Bangalore, for providing valuable assistance and advice. All those who kindly provided advice, suggestions, information and research resources for this study are duly acknowledged. Particular gratitude goes to Mr. Dileep D for his support throughout.

10 ix A very special appreciation is due to my wife Sudha S. K., not only for her constant encouragement but also for her patience and understanding throughout. I am also greatly indebted to my mother Rathna V. Mandi, my son Riteesh M. Mandi and my daughter Divya M. Mandi for giving me their love and encouragement throughout.

11 x CONTENTS Declaration. Abstract.. List of Tables.. List of Figures List of Symbols.. List of Abbreviations.. iii v xviii xlii xlv xlviii 1. Introduction Code Sequences m -Sequences Gold Sequences Kasami Sequences Udaya - Siddiqi Sequences Linear Recurrence Relation (LRR) Chaotic Sequences Proposed Scheme Generation of Discrete Chaotic Sequences over Z Generation of Chaotic Binary Sequences from Sequences over Z Chaotic Functions and Sequence over Finite Field GF(2 m ) Deriving Binary Sequence by Expressing Every Element in GF(2 m ) of the Sequence as Binary m Tuple Deriving Binary Sequence by Selecting a Particular Binary Bit from Each Element of the Sequence over GF(2 m ) Deriving Binary Sequence by Mapping Every Element in GF(2 m ) to GF(2) using Trace Function Correlation Bound Bit Error Rate Performance... 18

12 xi 1.8 Contributions of the Thesis Organization of the Thesis Literature Survey Chaotic Functions and Sequences Early Methods of Deriving Binary Sequences from Chaotic Sequences Applications of Chaotic Binary Sequences in DS-CDMA Summary Objectives of the Present Work Motivation Objectives of the Present Work Mathematical Preliminaries, Overview of Chaotic Functions and Properties of Chaotic Sequences Mathematical Preliminaries Groups Rings Fields Polynomial over GF(p) Polynomial Arithmetic Irreducible Polynomial Primitive Polynomial over GF(2) Binary Finite Fields Trace Function Hamming Cross Correlation (HCCR) of Sequences Linear Complexity of Sequences Logistic Map Equations Tent Map Equation Cubic Map Equation Quadratic Map Equation Bernoulli Map Equation Summary Generation of Chaotic Discrete and Binary Sequences Generation of Chaotic Discrete Sequences over Z

13 xii 5.2 Generation of Chaotic Binary Sequences Mapping P 0 Based on Binary Conversion Polynomial Mapping P 1, P 2 and P 3, from Z 4 to Binary Summary Generation of Binary Sequences from Proposed Chaotic Functions Defined over Finite Field GF(2 m ) Describing Chaotic Functions over Finite Field GF(2 m ) Generation of Chaotic Sequence over Finite Field GF(2 m ) Chaotic Sequence over Finite Field GF(2 8 ) Expressing Every Element in GF(2 8 ) of the Sequence as 64 Binary 8 Tuple Selecting a Particular Bit from Each Element of the Sequence over GF(2 8 ) Mapping Every Element in GF(2 8 ) to GF(2) using Trace Function Chaotic Sequence over Finite Field GF(2 16 ) Expressing Every Element in GF(2 16 ) of the Sequence as Binary 16 Tuple Selecting a Particular Bit from Each Element of the Sequence over GF(2 16 ) Mapping Every Element in GF(2 16 ) to GF(2) using Trace Function Summary Properties of Chaotic Discrete Sequences Over Z 4 and Corresponding Binary Sequences Derived from them Hamming CCR (HCCR) and Balance Properties of Segments of Chaotic Discrete Sequences over Z HCCR and Balance Properties of Discrete Sequences over Z 4 Generated using Logistic Map Equations HCCR and Balance Properties of Discrete Sequences over Z 4 Generated using Tent Map Equation HCCR and Balance Properties of Discrete Sequences over Z 4 Generated using Cubic Map Equation.. 77

14 xiii HCCR and Balance Properties of Discrete Sequences over Z 4 Generated using Quadratic Map Equation HCCR and Balance Properties of Discrete Sequences over Z 4 Generated using Bernoulli Map Equation CCR Properties of Chaotic Binary Sequences Derived from Sequence over Z 4 using Mapping P 0 (y) Linear Complexity Properties of Segments of Binary Sequences of Length 15 bits Derived from Sequence Over Z 4 using Mapping P LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Mapping P LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map 97 Equation and Mapping P LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map 100 Equation and Mapping P LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Mapping P LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Mapping P CCR Properties of Chaotic Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping Linear Complexity Properties of Segments of Binary Sequences of Length 15 bits Derived from Sequence Over Z 4 using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map

15 xiv Equations and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map 122 Equations and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equations and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equations and Using Polynomial Mapping CCR Properties of Chaotic Binary Sequences Derived From Sequence over Z 4 Using Polynomial Mapping Linear Complexity Properties of Segments of Binary Sequences of Length 15 bits Derived from Sequence Over Z 4 using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map Equation and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map 144 Equation and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Using Polynomial Mapping

16 xv 7.8 CCR Properties of Chaotic Binary Sequences Derived From Sequence over Z 4 Using Polynomial Mapping Linear Complexity Properties of Segments of Binary Sequences of Length 15 bits Derived from Sequence Over Z 4 using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map Equation and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map Equation and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Using Polynomial Mapping LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Using Polynomial Mapping Results and Discussion Summary Properties of Chaotic Discrete Sequences Over GF(2 m ) and Corresponding Binary Sequences Derived from them Hamming Autocorrelation (HACR) and Balance Properties of Sequences over GF(2 m ) HACR Properties of Sequences over GF(2 8 ) Balance Properties of Sequences over GF(2 8 ) CCR Properties of Segments of Binary Sequences Derived from Chaotic Sequence over GF(2 8 )

17 xvi CCR Properties of Binary Sequences Generated by Expressing Field Element in GF(2 8 ) as Binary 8 Tuple CCR Properties of Binary Sequences Generated by Selecting a Particular Bit from Each Element of the Sequence Over GF(2 8 ) CCR Properties of Binary Sequences Generated by Mapping Every Field Element in GF(2 8 ) to GF(2) using Trace Function LC Properties of Segments of Binary Sequences of length 15 bits Derived from Chaotic Sequence over GF(2 8 ) LC Properties of Binary Sequences of length 15 bits Generated by Expressing Field Element in GF(2 8 ) as Binary 8 Tuple LC Properties of Binary Sequences of length 15 bits Generated by Selecting a Particular Bit from Each Element of the Sequence Over GF(2 8 ) LC Properties of Binary Sequences of length 15 bits Generated by Mapping Every Field Element in GF(2 8 ) to GF(2) using Trace Function Hamming Autocorrelation (HACR) and Balance Properties of Sequences over GF(2 16 ) HACR Properties of Sequences over GF(2 16 ) Balance Properties of Sequences over GF(2 16 ) CCR Properties of Segments of Binary Sequences Derived from Chaotic Sequence over GF(2 16 ) CCR Properties of Binary Sequences Generated by Expressing Field Element in GF(2 16 ) as Binary 16 Tuples CCR Properties of Binary Sequences Generated by Selecting a Particular Bit from Each Element of the Sequence Over GF(2 16 ).. 210

18 xvii CCR Properties of Binary Sequences Generated by Mapping Every Field Element in GF(2 16 ) to GF(2) using Trace Function LC Properties of Segments of Binary Sequences of length 15 bits Derived from Chaotic Sequence over GF(2 16 ) LC Properties of Binary Sequences of length 15 bits Generated by Expressing Field Element in GF(2 16 ) as Binary 16 Tuple LC Properties of Binary Sequences of length 15 bits Generated by Selecting a Particular Bit from Each Element of the Sequence Over GF(2 16 ) LC Properties of Binary Sequences Generated by Mapping Every Field Element in GF(2 16 ) to GF(2) using Trace Function Results and Discussion Summary Correlation Bound and Performance Evaluation Correlation Bound Welch Bound Sidelnikov Bound Performance Evaluation Based on Peak Magnitude of Pairwise CCR Value Performance Evaluation Based on Mean Square Cross Correlation Values Summary Conclusion and Scope for Future Work Conclusion Future Scope Appendix Equation for Average Bit Error Rate References Publication Details Vitae

19 xviii LIST OF TABLES Table No. Description Page No. Table 1.1 Details of Non-Overlapping Segments 12 Table 1.2 Parameters Considered for Generating Chaotic Binary Sequences. 13 Table 4.1 List of Irreducible Polynomials Over GF(2) Table 5.1 Polynomial Mappings P i (y), i = 1, 2, 3 59 Table 6.1 Some Mapping using Trace Function for Elements in GF(2 8 ). 66 Table 6.2 Sequence Generated in Example (6.1).. 67 Table 6.3 Sequence Generated in Example (6.2).. 68 Table 7.1 Hamming Cross Correlation values of Segments of Discrete Sequence of Length 128 Generated using Logistic Map given by Equation (7.1) 73 Table 7.2 Hamming Cross Correlation values of Segments of Discrete Sequence of Length 128 Generated using Logistic Map given by Equation (7.2) 75 Table 7.3 Hamming Cross Correlation values of Segments of Discrete Sequence of Length 128 Generated using Tent Map given by Equation (7.3) 76 Table 7.4 Hamming Cross Correlation values of Segments of Discrete Sequence of Length 128 Generated using Cubic Map given by Equation (7.4) 78 Table 7.5 Hamming Cross Correlation values of Segments of Segments of Discrete Sequence of Length 128 of Length 128 Generated using Quadratic Map given by Equation (7.5) 79

20 xix Table 7.6 Table 7.7 Table 7.8 Table 7.9 Table 7.10 Table 7.11 Table 7.12 Table 7.13 Table 7.14 Hamming Cross Correlation Values of Segments of Segments of Discrete Sequence of Length 128 Generated using Bernoulli Map given by Equation (7.6) 81 Length of Sequence, Maximum Value of Pairwise CCR Value α max, Number of Sequences γ and LC of Gold and Kasami Sequences. 84 Length of Segments and the Corresponding Maximum Pairwise CCR values Chosen.. 85 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value.. 86 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value 87 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value.. 88

21 xx Table 7.15 Table 7.16 Table 7.17 Table 7.18 Table 7.19 Table 7.20 Table 7.21 Table 7.22 Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Mapping P 0 89 Linear complexity of 7 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 15 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 94 Linear complexity of 28 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 16 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 96 Linear complexity of 24 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 97 Linear complexity of 9 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number... 98

22 xxi Table 7.23 Table 7.24 Table 7.25 Table 7.26 Table 7.27 Table 7.28 Table 7.29 Table 7.30 Linear complexity of 17 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number 99 Linear complexity of 24 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number 99 Linear complexity of 7 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 15 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 101 Linear complexity of 31 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 101 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 13 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Linear complexity of 30 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 104

23 xxii Table 7.31 Table 7.32 Table 7.33 Table 7.34 Table 7.35 Table 7.36 Table 7.37 Table 7.38 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 16 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 106 Linear complexity of 32 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 106 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value 109 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value 109 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.4) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value 110 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value 110 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value 110

24 xxiii Table 7.39 Table 7.40 Table 7.41 Table 7.42 Table 7.43 Table 7.44 Table 7.45 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value. 111 Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping P 1 (y) 111 Linear complexity of 9 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 115 Linear complexity of 19 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Linear complexity of 29 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 116 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 22 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number.. 118

25 xxiv Table 7.46 Table 7.47 Table 7.48 Table 7.49 Table 7.50 Table 7.51 Table 7.52 Linear complexity of 32 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 119 Linear complexity of 10 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 120 Linear complexity of 23 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Linear complexity of 31 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Linear complexity of 10 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 25 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 123 Linear complexity of 33 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 124

26 xxv Table 7.53 Table 7.54 Table 7.55 Table 7.56 Table 7.57 Table 7.58 Linear complexity of 9 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 126 Linear complexity of 23 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 126 Linear complexity of 32 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 127 Linear complexity of 7 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 20 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 129 Linear complexity of 37 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 129

27 xxvi Table 7.59 Table 7.60 Table 7.61 Table 7.62 Table 7.63 Table 7.64 Table 7.65 Table 7.66 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping P 2 (y) 134 Linear complexity of 11 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number.. 138

28 xxvii Table 7.67 Table 7.68 Table 7.69 Table 7.70 Table 7.71 Table 7.72 Table 7.73 Linear complexity of 21 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 138 Linear complexity of 31 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 139 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 25 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 141 Linear complexity of 32 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 141 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 143 Linear complexity of 22 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number.. 143

29 xxviii Table 7.74 Table 7.75 Table 7.76 Table 7.77 Table 7.78 Table 7.79 Linear complexity of 34 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 24 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 146 Linear complexity of 35 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 146 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 19 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 148

30 xxix Table 7.80 Table 7.81 Table 7.82 Table 7.83 Table 7.84 Table 7.85 Table 7.86 Linear complexity of 27 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 149 Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 21 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 151 Linear complexity of 28 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 151 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value.. 154

31 xxx Table 7.87 Table 7.88 Table 7.89 Table 7.90 Table 7.91 Table 7.92 Table 7.93 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping P 3 (y) 156 Linear complexity of 11 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear complexity of 21 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 160 Linear complexity of 30 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 161

32 xxxi Table 7.94 Linear complexity of 7 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Table 7.95 Linear complexity of 23 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 163 Table 7.96 Linear complexity of 34 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 163 Table 7.97 Linear complexity of 12 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 165 Table 7.98 Linear complexity of 23 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 165 Table 7.99 Linear complexity of 34 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 166 Table Linear complexity of 11 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number.. 168

33 xxxii Table Linear complexity of 23 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 168 Table Linear complexity of 26 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 169 Table Linear complexity of 7 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Table Linear complexity of 25 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 171 Table Linear complexity of 36 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 171 Table Linear complexity of 8 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number.. 173

34 xxxiii Table Linear complexity of 25 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 173 Table Linear complexity of 33 Binary Sequences of Length 15 Bits Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number. 174 Table Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z Table Comparison of peak magnitude of Pairwise CCR value of Proposed Sequences with Earlier Methods Table 8.1 Number of Segments out of 33 Segments Derived from Sequence over GF(2 8 ) defined by Logistic map Equation (7.10) and by Expressing Element in GF(2 8 ) as 8 tuple having Magnitude of Pairwise CCR Value 0.35, 0.5 and Table 8.2 Number of Segments out of 136 Segments Derived from Sequence over GF(2 8 ) defined by Logistic map Equation (7.11) and by Expressing Element in GF(2 8 ) as 8 tuple having Magnitude of Pairwise CCR Value 0.35, 0.5 and Table 8.3 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 8 ) by Expressing Element in GF(2 8 ) as 8 tuple 187

35 xxxiv Table 8.4 Table 8.5 Table 8.6 Table 8.7 Table 8.8 Table 8.9 Table 8.10 Number of Segments out of 4 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 8 ) of Length 15 having Magnitude of Pairwise CCR Value 0.35, 0.5 and Number of Segments out of 17 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Selecting a Particular Bit from Element Over GF(2 8 ) of Length 15 having Magnitude of Pairwise CCR Value 0.35, 0.5 and Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 8 ) by Selecting a Particular Bit from Element Over GF(2 8 ) 189 Number of Segments out of 4 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and Using Trace Function having Magnitude of Pairwise CCR Value 0.35, 0.5 and Number of Segments out of 17 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and Using Trace Function having Magnitude of Pairwise CCR Value 0.35, 0.5 and Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 8 ) using Trace Function. 192 Linear Complexity of 4 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number.. 194

36 xxxv Table 8.11 Table 8.12 Table 8.13 Table 8.14 Table 8.15 Table 8.16 Linear Complexity of 8 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 195 Linear Complexity of 19 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number 195 Linear Complexity of 2 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear Complexity of 5 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number. 196 Linear Complexity of 30 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number 197 Linear Complexity of 2 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number.. 198

37 xxxvi Table 8.17 Linear Complexity of 3 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Table 8.18 Linear Complexity of 4 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Table 8.19 Linear Complexity of 4 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Table 8.20 Linear Complexity of 11 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Table 8.21 Linear Complexity of 15 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Table 8.22 Linear Complexity of 2 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and using Trace Function having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 202 Table 8.23 Linear Complexity of 3 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and using Trace Function having Magnitude of Pairwise CCR Value 0.5 along with Segment Number.. 202

38 xxxvii Table 8.24 Linear Complexity of 3 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.10) and using Trace Function having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Table 8.25 Linear Complexity of 2 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and using Trace Function having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 203 Table 8.26 Linear Complexity of 6 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and using Trace Function having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Table 8.27 Linear Complexity of 12 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (7.11) and using Trace Function having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Table 8.28 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 16 ) as 16 Tuple having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g. 208 Table 8.29 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 16 ) as 16 Tuple having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g. 208 Table 8.30 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 16 ).. 209

39 xxxviii Table 8.31 Table 8.32 Table 8.33 Table 8.34 Table 8.35 Table 8.36 Table 8.37 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Different Magnitude of Pairwise CCR Value α with x 0 = g, r 1 = g and r 2 = g. 211 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (7.11) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Different Magnitude of Pairwise CCR Value α with x 0 = g, r 1 = g and r 2 = g Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 16 ) by Selecting a Particular Bit from Element Over GF(2 16 ) Number of Segments of Different Lengths Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.10) and using Trace Function having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g Number of Segments of Different Lengths Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.11) and using Trace Function having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 16 ) using Trace Function Linear Complexity of 7 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 218

40 xxxix Table 8.38 Table 8.39 Table 8.40 Table 8.41 Table 8.42 Table 8.43 Table 8.44 Table 8.45 Linear Complexity of 20 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Linear Complexity of 26 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.10) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Linear Complexity of 8 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number 221 Linear Complexity of 22 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Linear Complexity of 29 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (7.11) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Linear Complexity of 9 Segments Derived from sequence over GF(2 16 ) using Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Linear Complexity of 21 Segments Derived from sequence over GF(2 16 ) using Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number 224 Linear Complexity of 34 Segments Derived from sequence over GF(2 16 ) using Logistic Map Equation (7.10) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number 224

41 xl Table 8.46 Table 8.47 Table 8.48 Table 8.49 Table 8.50 Table 8.51 Table 8.52 Table 8.53 Linear Complexity of 9 Segments Derived using Logistic map Equation (7.11) by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Linear Complexity of 25 Segments Derived using Logistic map Equation (7.11) by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number 226 Linear Complexity of 32 Segments Derived using Logistic map Equation (7.11) by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number 227 Linear Complexity of 10 Binary Sequences Derived using Logistic Map Equation (7.10) and Trace Function having Magnitude of Pairwise CCR value 0.35 along with Segment Number. 229 Linear Complexity of 20 Binary Sequences Derived using Logistic Map Equation (7.10) and Trace Function having Magnitude of Pairwise CCR value 0.5 along with Segment Number. 229 Linear Complexity of 30 Binary Sequences Derived using Logistic Map Equation (7.10) and Trace Function having Magnitude of Pairwise CCR value 0.6 along with Segment Number. 230 Linear Complexity of 9 Binary Sequences Derived using Logistic Map Equation (7.11) and Trace Function having Magnitude of Pairwise CCR value 0.35 along with Segment Number. 231 Linear Complexity of 22 Binary Sequences Derived using Logistic Map Equation (7.11) and Trace Function having Magnitude of Pairwise CCR value 0.5 along with Segment Number. 232

42 xli Table 8.54 Linear Complexity of 29 Binary Sequences Derived using Logistic Map Equation (7.11) and Trace Function having Magnitude of Pairwise CCR value 0.6 along with Segment Number Table 8.55 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over Finite Field Table 9.1 Correlation Bound for Gold and Kasami Sequences 241 Table 9.2 Correlation Bound for Segments of Sequences derived from Logistic map Equation (7.1) and mapping P Table 9.3 Correlation Bound for Segments of Sequences derived from Logistic map Equation (7.2) and mapping P Table 9.4 Normalized Mean Square Cross Correlation Values ( of Proposed Segments of Sequences. 254 Table 9.5 Minimum values of for different lengths 257 Table 9.6 Bit Error Rate Comparison of Proposed Segments of Sequences with Earlier Methods. 258

43 xlii LIST OF FIGURES Figure No. Description Page No. Figure 1.1 Non-Overlapping Segments of Length 15 Bits.. 12 Figure 4.1 One-Point Attractor of Logistic Map given by Equation (4.11) Figure 4.2 N-point Attractor of Logistic Map given by Equation (4.11).. 46 Figure 4.3 Bifurcation Diagram of Logistic Map given by Equation (4.11). 47 Figure 4.4 Sensitivity to Initial Conditions of Logistic Map given by Equation (4.11) 48 Figure 4.5 Sensitivity to Initial Condition of Logistic Map given by Equation (4.12) 49 Figure 4.6 Sensitivity to Initial Condition of Tent Map given by Equation (4.13) 50 Figure 4.7 Sensitivity to Initial Condition of Cubic Map given by Equation (4.14) 51 Figure 4.8 Sensitivity to Initial Condition of Quadratic Map given by Equation (4.15) 52 Figure 4.9 Bifurcation Diagram of Quadratic Map given by Equation (4.15). 52 Figure 4.10 Sensitivity to Initial Condition of Bernoulli Map given by Equation (4.16) 53 Figure 4.11 Bifurcation Diagram of Bernoulli Map given by Equation (4.16). 54 Figure 5.1 Scheme of Generating Chaotic Discrete Sequence. 58 Figure 6.1 Block Diagram for Generation of Chaotic Sequence Over GF(2 m ). 63 Figure 7.1 Non-overlapping segments of length 128 of discrete sequence over Z

44 xliii Figure 7.2 Histogram of Discrete Sequence Generated using Logistic Map given by Equation (7.1) 74 Figure 7.3 Histogram of Discrete Sequence Generated using Logistic Map given by Equation (7.2) 75 Figure 7.4 Histogram of Discrete Sequence Generated using Tent Map given by Equation (7.3) 77 Figure 7.5 Histogram of Discrete Sequence Generated using Cubic Map given by Equation (7.4) 78 Figure 7.6 Histogram of Discrete Sequence Generated using Quadratic Map given by Equation (7.5) 80 Figure 7.7 Histogram of Discrete Sequence Generated using Bernoulli Map given by Equation (7.6) 82 Figure 9.1 Simplified Diagram of a CDMA System 243 Figure 9.2 Variation of BER with Number of Users for Sequences of Length 15 Bits and = 8 db Figure 9.3 Variation of BER with for Sequences of Length 15 Bits and Number of Simultaneous Users = Figure 9.4 Variation of BER with Number of Users for Sequences of Length 31 Bits and = 8 db Figure 9.5 Variation of BER with for Sequences of Length 15 Bits and Number of Simultaneous Users = Figure 9.6 Variation of BER with Number of Users for Sequences of Length 63 Bits = 8dB Figure 9.7 Variation of BER with for Sequences of Length 63 Bits and Number of Simultaneous Users = Figure 9.8 Variation of BER with Number of Users for Sequences of Length 127 Bits = 8dB 250 Figure 9.9 Variation of BER with for Sequences of Length 127 Bits and Number of Simultaneous Users =

45 xliv Figure 9.10 Variation of BER with Number of Users for Sequences of Length 255 Bits = 8dB Figure 9.11 Variation of BER with for Sequences of Length 255 Bits and 251 Number of Simultaneous Users = 60.. Figure 9.12 Variation of BER with Number of Users for Sequences of Length 15 and = 8 db using Mapping P Figure 9.13 Variation of BER with for Sequences of Length 15 and Number of Simultaneous Users equal to

46 xlv LIST OF SYMBOLS {x k } Chaotic real valued sequence {y k } Chaotic discrete sequence over Z 4 {b k } Chaotic binary sequence r, r 1, r 2 Bifurcation parameters x 0 Initial value Tr(β) Trace of an element β in GF(2 m ) n(τ) Number of agreements d(τ) Number of disagreements τ Number of locations by which one sequence is shifted w.r.t. other Z 4 Ring of residue class integers modulo 4 α Magnitude of pairwise cross correlation value γ Number of sequences n Multiplication factor, positive integer m Positive integer p Prime integer S j (t) Signal originating from user j P j m j (t) a j (t) T c T r(t) n(t) (j) Z i Φ j Received power of user j Data sequence for user j Spreading sequence Chip period Symbol period Received signal at the input of Receiver Additive White Gaussian Noise Correlator output for user j at the end of i th bit period Arbitrary phase of each carrier Multiple access interference of i th bit period Noise contribution at i th bit period

47 xlvi (j) I i ω 0 m j,i E b N 0 /2 Desired contribution from user j Carrier Frequency Data of the j th user at the end of i th bit period Signal energy per bit period Double sided white noise power spectral density Signal to noise ratio C j,k (l) K 2 N L s x k+1 x k Cross correlation function of two sequences Number of simultaneous users Mean square cross correlation value Normalized mean square cross correlation value Length of the spreading sequence Number of stages Segment length in bits Next Value Present Value P 0 (y) Binary Conversion of integers 0, 1, 2 and 3 P 1 (y) Polynomial mapping defined by 2y mod 4 P 2 (y) Polynomial mapping defined by (y 2 y) mod 4 P 3 (y) Polynomial mapping defined by (y 2 + y) mod 4 G Group R Ring F Field + Addition operation in G, R or F x Multiplication operation in G, R or F Є Belongs to {a, b, c, } Elements in a set e Unique identity element * Binary operation a i Unique inverse in a Group -a Additive inverse of element a in G a -1 Multiplicative inverse of element a in G

48 xlvii R xy (τ) Normalized Cyclic Hamming Cross Correlation function of two sequences x and y P e Average Bit Error Rate g Generating element or primitive element in GF(2 m ) Integer Part of u M Period of discrete sequence over GF(2 m ) Λ Number of segments of given length s

49 xlviii LIST OF ABBREVIATIONS AWGN Additive White Gaussian Noise BER Bit Error Rate BPSK Binary Phase Shift Keying BW Band Width CCR Cross Correlation CDMA Code Division Multiple Access CML Coupled Map Lattice db Decibel DS Direct Sequence DS-CDMA Direct Sequence Code Division Multiple Access DSSS Direct Sequence Spread Spectrum DT Discrete Time FDMA Frequency Division Multiple Access FH Frequency Hopping GF(2 m ) Finite Field of order 2 m HACR Hamming Autocorrelation HCCR Hamming Cross Correlation LC Linear Complexity LFSR Linear Feedback Shift Register MAI Multiple Access Interference MC Multi Carrier m-l Maximal Length MMSE Minimum Mean Square Error OSCM Over Sampled Chaotic Map PDF Probability Density Function PIC Parallel Interference Cancellation PN Pseudo Noise

50 xlix PSD PSK PWL QAM QoS SNR SS TDMA TH VHDL XOR Power Spectral Density Phase Shift Keying Piece Wise Linear Quadrature Amplitude Modulation Quality of Service Signal to Noise Ratio Spread Spectrum Time Division Multiple Access Time Hopping Very High speed integrated circuit Hardware Description Language Exclusive OR

51 List of Corrections listed below. The corrections with reference to the remarks of the Examiner are carried out as S. Remark of Existing Correction Modified Page No. External Examiner Page No. No. 1. Change hamming Page 19, hamming is changed to Page 19 and in to Hamming. You line (-2) Hamming. Page No s: 10, should change this 73, 74, 77, 78, error in other places 79, 82, 179, also Remove very from Page 23, The sentence is modified Page 23 the sentence. line 4 by removing the term very from the sentence. 3. Give a blank after Page 57, Sectin 1.3 is rewritten as Page 57 Section. line 13 Section I think you mean Page iv Knowledge of Fountain Page iv Fountain of is changed to Fountain Knowledge. of Knowledge. 5. Change to two small Page 20, The sentence is rewritten Page 20 sentences. line (-6) into two small sentences. Mahalinga V. Mandi (Reg. No: ECE04D004) Co-Supervisor: Dr. R. Murali. Supervisor: Dr. K.N. Hari Bhat

52 CHAPTER 1 INTRODUCTION

53 1 CHAPTER 1 INTRODUCTION One of the earliest secure communication technique used for military purpose was Spread Spectrum (SS) communication (Raymond L. Pickholtz et al 1982). Spread spectrum is a means of transmission in which the signal occupies a bandwidth in excess of the minimum necessary to send the information; the band spread is accomplished by means of a code sequence which is independent of the data and a synchronized reception with the code sequence at the receiver is used for despreading and subsequent data recovery. Hence in SS technique a signal is transmitted having a bandwidth considerably greater than the bandwidth of baseband signal. Some of the desirable features of SS communications are low probability of interception (LPI), immunity against jamming and separation of multipath signals (Raymond L. Pickholtz et al 1982). One of the widely used spread spectrum technique is direct sequence spread spectrum (DSSS). In DSSS system, the information signal is first modulated and then spread in bandwidth prior to transmission over the channel. At the receiver, the reveived signal is despread in bandwidth by the same amount and demodulated to recover the original information signal. The code sequence used for spreading is a noise like wideband sequence having good autocorrelation (ACR) property. The periodic autocorrelation function (ACF) is nearly two valued with peaks at zero shift and is zero elsewhere. One of the most commonly used code sequences in SS communication is m-sequences (Solomon W. Goulomb 1967) having good ACR property and is generated using linear feedback shift register (LFSR). By using a single shift-register, maximum length sequences can be created and are called m-sequences. The normalized ACR value of m-sequence is two valued and is equal to 1 (for no shift) and -1/(2 L -1) for all other shifts, where L is the number of stages of LFSR.

54 2 By having a set of wideband noise like code sequences having not only good ACR values but also having pairwise low cross correlation (CCR) values, it is possible to have simultaneous transmission of different signals, all occupying the same bandwidth. At the receiver the signals are separated by cross correlating the received demodulated sequence with corresponding reference code sequence. This is possible because the cross correlations between the code sequence of the desired user and the code sequences of the other users are small. This scheme is called Direct Sequence Code Division Multiple Access (DS-CDMA) (Raymond L. Pickholtz et al 1982). In DS-CDMA, a pseudorandom sequence having a higher bandwidth than the information signal is used to spread the information signal directly. The resultant signal has a significantly higher bandwidth than the original signal. Examples of set of sequences commonly used in DS-CDMA are Gold sequences and Kasami sequences (Byeong Gi Lee and Byoung-Hoon Kim 2002). Gold sequences are generated by bit by bit modulo-2 addition of the two maximum length sequences generated by using two distinct LFSR of same number of stages. A procedure similar to that used for generating Gold sequences can generate binary sequences called Kasami sequences. The performance of binary communication systems is expressed interms of Bit Error Rate (BER). It gives the average number of detected binary bits in error at the receiver in a fairly long binary sequence. In a DS-CDMA system the average BER performance depends on the bit energy, white noise power spectral density (PSD), number of simultaneous users and the normalized mean square value of cross correlation of the spreading sequences assigned to the users. There is a fundamental limit on how the correlation can be for a family of sequences. The Welch bound (Welch L.R. 1974) and Sidelnikov bound (Sidelnikov V.M. 1971) provide lower bounds on non-trivial peak magnitude cross correlations values denoted by which is considered as an important performance factor of CDMA systems (Massey J.L. 1991). Gold sequences do not achieve Welch lower bound and Kasami sequences (small set) satisfy Welch lower bound (John G. Proakis 2001). Udaya

55 3 - Siddiqi (US) sequences satisfy both Welch bound and Sidelnikov bound with equality (Udaya P. and Siddiqi M.U. 1996). Linear complexity (LC) of a periodic sequence is the length of the shortest linear feedback shift register that can be used to generate the sequence (Massey J.L. 1969). Spreading sequences having larger linear complexity is another important parameter to be considered. The necessary security to the CDMA systems can be achieved by having sequences with large linear complexity in which case a large segment of code sequence is to be known to get complete code sequence. In the next section, desirable properties of code sequences are considered. 1.1 CODE SEQUENCES In this section, the properties of m-sequences, Gold Sequences, Kasami Sequences, Udaya Siddiqi (US) sequences and Chaotic sequences are considered. The properties of m sequences (Solomon W. Goulomb 1967) is considered first m-sequences Periodic binary sequences can be generated by using shift registers with appropriate linear feedback. Consider an L stage shift register with linear feedback such that the period of the sequence N is maximum possible and is equal to (2 L -1). Such sequences are called m-sequences. They exhibit following properties. i) Balance property: There are (2 L - 1 ), 1 s and (2 L - 1-1), 0 s ii) Autocorrelation (ACR): Normalized ACR (NACR) is two valued and is equal to 1 and -1/(2 L -1) iii) Run property: The number of runs (either zeros or ones) of various lengths in the sequence is as expected for a random sequence

56 4 iv) Linear Complexity: L, equal to the number of stages v) Number of distinct sequences of period (2 L -1) is equal to number of primitive polynomials of degree L over GF(2). With increase in number of stages L, period N also increases and off-peak normalized autocorrelation value approaches zero. Therefore, m-sequences posses desirable autocorrelation values. But the periodic cross correlation (CCR) values between any pair of m-sequences of the same period has relatively large peaks. As the number of m-sequences increases rapidly with the number of stage L, the probability of large cross correlation peaks becomes high for long m-sequences (Sarwate D. V. and Pursley M. B. 1980). Hence such high values of cross correlation are not desirable in the CDMA communications. The properties of Gold sequences are considered next Gold Sequences Sequences with better periodic cross correlation (CCR) properties were derived from m-sequences by Gold (Robert Gold 1967 and 1968). Gold proved that certain pairs of m-sequences of length N equal to (2 L 1) exhibit a three-valued cross correlation function with the values {-1, - t(l), t(l) 2} where, (1.1) Two m-sequences of length N that yield a periodic cross-correlation function taking the possible values {-1, - t(l), t(l) 2}are called preferred pairs or preferred sequences. There are γ equal to (2 L + 1) Gold sequences of length N which are generated by bit by bit modulo-2 addition of the two periodic sequences generated by using two LFSR of L stages. (2 L -1) sequences have linear complexity equal to 2L and two sequences have linear complexity equal to L (Key E.L. 1976). Similar to the case of the cross-correlation function, the off-peak autocorrelation function for a Gold sequence

57 5 takes the values in the set {-1, - t(l), t(l) 2}. Gold sequences are optimal with respect to Sidelnikov bound if the number of stages is odd and suboptimal with respect to Sidelnikov bound if the number of stages is even (Udaya P. and Siddiqi M.U. 1996). The properties of Kasami sequences are considered next Kasami Sequences A procedure similar to that used for generating Gold sequences can generate a smaller set of binary sequences called small set Kasami sequences (Kasami T. 1966) of period N equal to (2 L 1), for an even L. Kasami sequences do not exist for odd number of stages L and hence there are no Kasami sequences of lengths 31 for L equal to 5, 127 for L equal to 7 and so on. The autocorrelation function as well as the cross-correlation function of small set Kasami sequences takes a value in the set {-1, - (2 L/2 +1), (2 L/2-1)}. Small set Kasami sequences have better cross correlation properties compared to Gold sequences (Sarwate D. V. et al 1980). However, the small set Kasami sequences contains number of sequences γ equal to 2 L/2. Large set Kasami sequences contains number of sequences γ equal to 2 L/2 (2 L +1) and the cross correlation function is five valued given by {-1, -(2 (L+2)/2 +1), (2 (L+2)/2-1), -(2 L/2 +1), (2 L/2-1)}. So the maximum magnitude of pairwise cross correlation value denoted by α max of Kasami sequences (large set) remains same as that of the Gold sequences. Also the LC of large set Kasami sequences is more than that of Gold sequences and the LC of small set Kasami sequences is less than that of Gold sequences (Andrew Klapper 1995). Kasami sequences (small set) are optimal with respect to Welch bound for even number of stages. However, Kasami sequences (large set) are suboptimal with respect to Sidelnikov bound for even number of stages (Udaya P. and Siddiqi M.U. 1996). The properties of Udaya Siddiqi sequences are considered next.

58 Udaya - Siddiqi Sequences Udaya Siddiqi (Udaya P. and Siddiqi M.U. 1996) gave a method of constructing families of (2 r ) biphase sequences of period equal to 2(2 r 1), where r is a positive integer, derived from interleaved maximal-length families over Z 4. The biphase families are constructed from families of Z 4 sequences using polynomial mapping. These families satisfy Sidelnikov bound and Welch bound with eqaulity. The linear recurrence relation and their properties are considered next Linear Recurrence Relation (LRR) Consider recurrence relation of the form, (1.2) Where are the j coefficients and are j initial values., are all from real field or complex field or finite field. Then the Equation (1.2) is called linear recurrence relation (LRR) of order j and is over real field or finite field, if, are from real field or finite field respectively (Rudolf Lidl and Herald Niederriter 1984). If the constant term, it is called homogeneous LRR and if then it is called inhomogeneous LRR. With the j initial values and coefficients known, then for any can be determined recursively interms of its immediate past j value Chaotic Sequences A chaotic map is essentially an equation which is used iteratively to obtain a sequence starting with any initial value. Its general form is ); where k = 0, 1, 2, and x 0 being initial value. F is generally a non-linear function; k can be regarded as

59 7 time index, representing discrete instants of time and x k+1 depends only on previous value x k. Hence chaotic map can be regarded as a first order non-linear, time discrete, dynamical equation used iteratively to generate chaotic sequence x 0, x 1, x 2 and so on (Alligood K.T. et al 1997). Chaos is characterized by deterministic, nonlinear, non-periodic, non-converging and bounded behavior. Chaotic sequence is an example of discrete time continuous amplitude random sequence. Amplitude is generally bounded within 0 and 1 or -1 and 1 or -0.5 and 0.5. Chaotic sequences are characterized by very low cross correlation properties. The main characteristic of chaotic sequences is the sensitive dependence on initial conditions. Even though a small difference is introduced between the initial values, the two chaotic sequences separate rapidly from each other after a short time period and are highly uncorrelated (Robert M. May 1976). Therefore, by using different initial values, it is possible to produce a large number of chaotic sequences which are pairwise highly uncorrelated. One of the well known one-dimensional iterative maps which exhibit chaotic properties is the Logistic Map (Robert M. May 1976). The other chaotic maps which are considered in this work are Tent map (Michael Crampin et al 1994), Cubic map (Robert M. May 1984), Quadratic map (Morton P. 1998) and Bernoulli map (Robert M. May 1976). i) Logistic Map equations Logistic Map (1):, defined over real and 0 < x < 1 (1.3) where r is called as the bifurcation parameter or control parameter. For r values in the range 3.57 < r < 4, the discrete sequence {x k } is chaotic.

60 8 Logistic Map (2): where -1 < x < 1 (1.4) Here the bifurcation parameter r is chosen to be 1.72 < r < 2, which yield chaotic sequences. ii) Tent Map equation where 0 < x < 1 (1.5) iii) Cubic Map equation where -1 < x < 1 (1.6) iv) Quadratic Map equation where < x < 0.5 (1.7) Here the bifurcation parameter r is chosen to be 0.36 < r < 0.5 to get chaotic sequences. v) Bernoulli Map equation where < x < 0.5 (1.8) Here the bifurcation parameter r is chosen to be 1.2 < r < 2 to get chaotic sequences. Equations (1.3) to (1.8) can be regarded as first order non linear recurrence relation (NLRR) over real field. Further in Equations (1.4), (1.5), (1.7) and (1.8) constant term is not zero. For the choice of bifurcation parameter r, with in the range given along with the Equation (1.3), (1.4), (1.7) and (1.8), the sequence produced is non-periodic and

61 9 non-converging. Even with two initial values differing by a very small value, the resulting sequences are highly uncorrelated (Robert M. May 1976). In this work, methods of generating discrete and binary sequences using chaotic map equations are proposed. Their linear complexity and cross correlation properties are investigated. 1.2 PROPOSED SCHEME Using chaotic functions defined in earlier section, continous amplitude discrete time chaotic sequences are generated. In this work methods are proposed to derive discrete chaotic sequences and chaotic binary sequences from them. The following steps are involved. i) Generation of chaotic discrete sequences using multiplication of elements by integer n and reducing to integer modulo m, where n > m. In this work six values for n, n = 5, 6, 7, 8, 9, 10 and m = 4 is chosen. The sequence elements will have 4 discrete values, 0, 1, 2 and 3. ii) Generation of chaotic binary sequences derived from discrete sequence over Z 4. Using Binary representation of 0, 1, 2 and 3 in Z 4 by two bits Using three Polynomial mappings where x Є Z 4 is mapped to {0, 1} iii) Two chaotic functions over finite fields are defined. Generation of chaotic sequences over GF(2 8 ) and GF(2 16 ) using the two chaotic functions are discussed. Three methods are proposed to transform chaotic sequences over GF(2 m ) to binary for two standard word lengths m = 8 and m = 16. They are,

62 10 Expressing every element in GF(2 m ) of the sequence as binary m tuple Selecting a particular binary bit from each element of the sequence over GF(2 m ) Mapping every element in GF(2 m ) to GF(2) using Trace function 1.3 GENERATION OF DISCRETE CHAOTIC SEQUENCES OVER Z 4 Method of generation of discrete chaotic sequences using chaotic map equations is discussed in this section. For any chaotic function, by proper choice of bifurcation parameter r as specified, any initial value x 0, iteratively generates a real valued infinite sequence {x i }, i = 0, 1, 2,. where with x 0 in the specified range depending on the chaotic function, makes each x k to lie in the same range. Each element x k of sequence {x i } is then multiplied by an integer n and the fraction part is discarded. The integer part is then reduced to small integer y k modulo 4. It is necessary that n > 4. In this work for six values of n namely n = 5, 6, 7, 8, 9 and 10, discrete sequences {y k } over Z 4 are generated using all the six chaotic functions. Their Hamming CCR and balance properties are investigated. The detailed explanation is presented in Chapter 5. It is shown in Chapter 7, that for choice of multiplication factor, n = 5, 6, 7, 8, 9 and 10, it is possible to generate discrete sequences exhibiting good normalized Hamming cross correlation and balance properties. Methods of deriving binary sequences from discrete chaotic sequences over Z 4 are proposed next. The sequences are generated using all the chaotic functions defined by Equations (1.3) to (1.8). 1.4 GENERATION OF CHAOTIC BINARY SEQUENCES FROM SEQUENCES OVER Z 4 As discussed in Section 1.3, discrete sequence {y k } over Z 4 is generated for different choices of n = 5, 6, 7, 8, 9 and 10 for different initial values for all the six chaotic functions considered. The discrete sequence {y k } over Z 4 is mapped to binary using following mappings.

63 11 i) Mapping P 0 based on binary conversion Let y Є Z 4, then P 0 (y) = b 1 b 0, where b 1, b 0 Є {0,1}, such that b b = y k. The mapping P 0 is essentially representation of elements of Z 4 namely 0, 1, 2 and 3 in binary P 0 (0)=00, P 0 (1)=01, P 0 (2)=10 and P 0 (3)=11. ii) Polynomial mapping from Z 4 to binary (Udaya P and Siddiqi M.U. 1996) a) P 1 (y) = 2y mod 4 (1.9) P 1 : y Є Z 4 is mapped to {0, 2} and {0, 2} is further mapped to (1, -1) P 1 can also be regarded as a mapping y Є Z 4 mapped to and then -1 is regarded as 0. b) P 2 (y) = (y 2 y) mod 4 (1.10) P 2 : y Є Z 4 is mapped to {0, 2} and {0, 2} is further mapped to (1, -1) P 2 can also be regarded as a mapping y Є Z 4 mapped to and -1 is regarded as 0. c) P 3 (y) = (y 2 + y) mod 4 (1.11) P 3 : y Є Z 4 is mapped to {0, 2} and {0, 2} is further mapped to (1, -1) P 3 can also be regarded as a mapping y Є Z 4 mapped to and -1 is regarded as 0. A sequence of length N over Z 4 with mapping P 0 gives rise to a binary sequence of length 2N, whereas with mapping P 1 or P 2 or P 3 the resulting binary sequence is of length N. To study the linear complexity and cross correlation properties of the binary sequence, the generated binary sequence is divided into non-overlapping segments. The segments of sequences can be of any length. However, for the sake of comparison of results obtained in this work with earlier published works, sequences of length s equal to

64 12 15, 31, 63, 127 and 255 are considered. Details of the non-overlapping segments considered in this work are given in Table 1.1. Table 1.1 Details of Non-Overlapping Segments Length of sequence in bits Length of segment in bits s Number of segments Λ As an example, first 1500 bits (b 0 to b 1499 as shown in Fig. 1.1) of the generated binary sequence are considered which is divided into 15 bit non-overlapping segments. The number of segments of 15 bit that we get is 100. Each of these segments is numbered as Segment 1, Segment 2 and is shown in Fig The LC and pairwise normalized CCR values of these 100 segments are computed. BinarySequence b 0, b1,... b14, b15, b16,... b 29,... b1485,... b1499 Segment1 Segment 2 Segment100 Figure 1.1 Non-overlapping segments of length 15. Table 1.2 lists the parameters considered for generating the chaotic binary sequences using mapping P 0 and polynomial mapping P 1, P 2 and P 3 for all the six chaotic map equations. The segments of sequences of length 15, 31, 63, 127 and 255 are considered. First column lists the range of n values, second column gives the binary mapping and third column specifies the segment length s, along with number Λ of segments denoted by (s, Λ).

65 13 Table 1.2 Parameters Considered for Generating Chaotic Binary Sequences n value Binary mapping Segments length, Number of segments (s, Λ) 5, 6, 7, 8, 9 and 10 Binary Conversion P 0 (y), Polynomial mapping P 1 (y), Polynomial mapping P 2 (y) and Polynomial mapping P 3 (y) (15,100), (31, 100), (63, 200), (127, 300) and (255, 700) The linear complexities of binary sequences of different lengths are computed using Berlekamp Massey algorithm (Massey J.L. 1969) and pairwise cross correlation values are computed and compared with Gold sequences and large set Kasami sequences of same length. It is found that using the proposed scheme of deriving segments of binary sequences from chaotic sequences using P 0, P 1, P 2 and P 3, it is possible to obtain some segments of sequences of same length s, for s equal to 15, 31, 63, 127 and 255 having LC greater than that of Gold sequences and maximum magnitude of pairwise CCR value, less than that of Gold sequences. Also the results of segments of sequences of length 63 and 255, exhibit large LC as compared to large set Kasami sequences and there are segments of sequences whose maximum magnitude of pairwise CCR value less than that of large set Kasami sequences. The detailed results are tabulated in Chapter 7. Having discussed generation of discrete sequence over Z 4 and chaotic binary sequence derived from sequence over Z 4, chaotic functions over Finite Field GF(2 m ) is proposed in the next section.

66 CHAOTIC FUNCTIONS AND SEQUENCE OVER FINITE FIELD GF(2 m ) The concept of chaotic map equations over reals as discussed in Section 1.3 is extended to chaotic map equation defined over finite field GF(2 m ) and the properties of binary sequences derived from them are investigated in this work. Proposed chaotic map equations over GF(2 m ) are based on Logistic map defined by equation, (1.12) Where r 1, r 2 and x k Є GF(2 m ). Comparing this equation with linear recurrence relation (LRR) over finite fields given in Rudolf Lidl and Herald Niederriter (1984), it is seen that the chaotic function given by Equation (1.12) can be regarded as a first order non linear recurrence relation (NLRR). Theoretically, Equation (1.12) can be defined for m 1. However, investigation is carried out for two standard word lengths m = 8 and m =16. In both cases the period of the sequences for random choice of r 1 and r 2 is less than (2 m -1). From computer simulation it is found that the maximum period obtained for m = 8 is 63 and for m = 16 is 255. For proper choice of r 1, r 2 and x 0 satisfying Equation (1.12), a periodic sequence is generated. A variation of Logistic map defined by Equation (1.12) is proposed and used in this work and is given by, (1.13) Where r 1, r 2 and x k Є GF(2 m ). Equation (1.13) is a first order inhomogeneous linear recurrence relation (LRR) over GF(2 m ). Theoretically, Equation (1.13) can be defined for m 1. However in this work two standard word lengths, m = 8 and m = 16 are considered. It is seen that for 5284 random choices of r 1, r 2 and x 0, the maximum period of chaotic sequence over GF(2 8 ) is 255 and over GF(2 16 ) is

67 15 Using Equations (1.12) and (1.13), the random chaotic sequence over GF(2 m ) are mapped to binary using three different techniques namely, i) Expressing every element in GF(2 m ) of the sequence as binary m tuple ii) Selecting a particular binary bit from each element of the sequence over GF(2 m ) iii) Mapping every element in GF(2 m ) to GF(2) using the Trace equation Where trace function, denoted by Tr(β) of an element β Є GF(2 m ) is defined by, = (1.14) As stated earlier the standard word lenghts m = 8 and m = 16 are considered in this work. With m = 8, only non-overlapping segments of length 15 are considered and their CCR and LC properties are investigated. For m = 16, the investigation is done for non-overlapping segments of same length s, for s = 15, 31, 63, 127 and 255. The number of segments considered for each s is as listed earlier in Table Deriving Binary Sequence by Expressing Every Element in GF(2 m ) of the Sequence as Binary m Tuple For arbitrarily chosen values of x 0, r 1 and r 2, random sequence of finite field elements are generated for the Logistic map Equations (1.12) and (1.13). Each of these finite field elements are expressed as binary m tuple and concatenated to get binary sequence. The maximum period of the resulting binary sequence is m times the period of sequence over GF(2 m ). The maximum period of the binary sequence is 504 using Equation (1.12) and 2040 using Equation (1.13) for m = 8. Like wise for m = 16, the maximum period of the binary sequence is 4080 using Equation (1.12) and using Equation (1.13). The

68 16 binary sequence is then divided into non-overlapping segments of same length s, for s = 15, 31, 63, 127 and 255 with number of segments as given in Table 1.1. The linear complexity and cross correlation properties of the binary sequences generated using the above technique are investigated Deriving Binary Sequence by Selecting a Particular Binary Bit from Each Element of the Sequence Over GF(2 m ) In this method of binary mapping, with arbitrarily chosen values of x 0, r 1 and r 2 for the Equations (1.12) and (1.13), chaotic sequence over GF(2 m ) for m = 8 and m = 16, with maximum possible period is generated. Each of these elements over GF(2 m ) are expressed as binary m tuple. First bit (MSB) of the resulting binary m tuples are selected and concatenated. In the same way 2 nd bit to m th bit (LSB) are selected and corresponding bits are concatenated. This binary sequence is divided into nonoverlapping segments of same length s, for s = 15, 31, 63, 127 and 255 with number of such segments as given in Table 1.1. Their cross correlation and linear complexity (LC) properties are investigated Deriving Binary Sequence by Mapping Every Element in GF(2 m ) to GF(2) Using Trace Function The trace function (Solomon W. Coulomb et al 2005, Da Rocha Jr. V.C. 2006) denoted by Tr(β) of an element β Є GF(2 m ), relative to GF(2) is defined by the Equation (1.14) which is given below. = (1.14) Here each element in GF(2 m ), is mapped to one bit binary and are concatenated to get binary sequence. In this work, the binary sequence is divided into non-overlapping

69 17 segments of same length s, for s equal to 15, 31, 63, 127 and 255, are considered. The LC and cross correlation properties of these segments are investigated. In all the above cases, the properties of the segments of sequences are compared with Gold and large set Kasami sequences of same length. It is found that linear complexity of some segments of binary sequences is larger than that of Gold sequences and large set Kasami sequences. The number of segments of a given length having maximum magnitude of pairwise CCR values same as that of Gold sequences is more than the number of Gold sequences. However this number is less than the number of large set Kasami sequences. Also there are some segments of binary sequences whose maximum magnitude of pairwise CCR value is less than that of Gold sequences and large set Kasami sequences of same length. The correlation bond is considered in the next section. 1.6 CORRELATION BOUND There is a fundamental limit on how the correlation can be for a family of sequences. The Welch bound (Welch L.R. 1974) and Sidelnikov bound (Sidelnikov V.M. 1971) provide lower bounds on maximum non-trivial correlations denoted by which is considered as an important performance factor (Massey J.L. 1991). It is found that there is a smaller set of proposed segments of binary sequences derived from chaotic sequences with much lower than CCR values of Gold and large set Kasami sequences, satisfies the Welch bound and Sidelnikov bound better than Gold and large set Kasami sequences of same length. Application of these bounds on the CCR values of the binary sequences generated using proposed scheme is discussed in Chapter 9. In a binary data transmission scheme, the performance is evaluated in terms of bit error rate (BER). It is the number of bits received in error in a long received sequence expressed as a ratio. In general BER depends on bit energy and channel white noise power spectral density. In CDMA applications, BER also depends on the mean square

70 18 value of cross correlation of code sequences and the number of simultaneous users. Demodulated and detected binary digits at the receiver may be in error due to channel noise and intereference from communication signal from other users. BER is expressed interms of probability of bit error which is a ratio of average number of bits in error in a given length of a sequence. BER performance of the proposed sequences in CDMA application is discussed in the next section. 1.7 BIT ERROR RATE PERFORMANCE For a synchronous system that assumes synchronized spreading sequences and constant power levels for all K users, the average BER depends on the white noise Power Spectral Density (PSD) and Multiple Access Interference (MAI) power. The average BER, Pe is given by (Vladeanu C. et al 2001), (1.15) Where E b = P j T is the signal energy per bit period, N 0 /2 is the double sided white noise power spectral density, is the signal-to-noise ratio, K is number of users, is the mean square cross correlation value, N is the length of the spreading sequence and modulation is binary phase shift keying (BPSK). Since α is considered to be normalized magnitude of pairwise CCR value, substituting = α max 2 in Equation (1.15), gives bit error rate under worst conditions. Therefore Equation (1.15) can be written as, (1.16)

71 19 BER performance is invesitgated for the binary sequences derived from chaotic sequences using six chaotic map equations defined over reals and two chaotic map equations defined over GF(2 8 ) and GF(2 16 ). Probability of bit error of the proposed sequences are computed for worst case by considering peak magnitude pairwise CCR valu eand normalized mean square cross correlation value ( ) and compared with Gold sequences and large set Kasami sequences. It is found that as number of user s increases, BER also increases for a fixed value of. By using sequences generated by proposed scheme as code sequences having magnitude of pairwise CCR value less than that of Gold sequences, BER is less. Thus the rate of increase of probability of bit error with number of users is small if proposed sequences with low CCR values are used. The number of users in this case is limited by number of code sequences available. On the other hand for a BER less than or equal to a particular value, when the number of users are less, by using proposed binary sequences the transmitter power can be reduced. Detailed results of BER performance is discussed in Chapter 9. The contributions of the thesis are given in the next section. 1.8 CONTRIBUTION OF THE THESIS In this work six chaotic map equations such as Logistic map, Tent Map, Cubic Map, Quadratic Map and Bernoulli Map are considered. A method is proposed to generate chaotic discrete sequence by multiplying each element of chaotic sequence by an integer n and reducing the integer part to modulo 4. Discrete chaotic sequences over Z 4 are generated for arbitrarily chosen values of n equal to 5, 6, 7, 8, 9 and m equal to 4 and investigated for cross correlation and balance properties. It is shown that, the discrete sequences over Z 4 are found to have low Hamming CCR value, long period and good balance properties.

72 20 Four methods are proposed to derive binary sequences from discrete chaotic sequences over Z 4 and using four mappings P 0, P 1, P 2 and P 3 as discussed in Section 1.4. The linear complexity and cross correlation properties of chaotic binary sequences so obtained are investigated and compared with binary sequences derived from chaotic sequences using other techniques available in the literature and also with the conventional Gold and large set Kasami sequences. It is found that the linear complexity is high and peak CCR value is lesser than the binary sequences generated using other techniques and conventional Gold and large set Kasami sequences. Another method is proposed to generate chaotic sequences from chaotic functions defined over finite field GF(2 m ). In this work, two standard values of m equal to 8 and 16 and two logistic maps are considered. One of the two chaotic maps can be regarded as first order non linear recurrence relation (NLRR) and the other chaotic map can be regarded as first order inhomogeneous linear recurrence relation (LRR). The sequences generated by using them have properties different from sequences generated using first order homogeneous linear recurrence relation over finite field. Sequences over GF(2 8 ) and GF(2 16 ) are transformed to binary using three techniques, i) Expressing field elements as binary m tuple. ii) Choosing bits in location i (where i = 1 to m) of each symbol in the sequence. iii) Mapping into binary using Trace function, where every field element in GF(2 m ) is mapped to 0 or 1 in GF(2). The cross correlation and linear complexiy properties of the proposed sequences are investigated. The results are compared with those of Gold and large set Kasami sequences. In all the above cases, it is found that, it is possible to generate more number of segements of sequences with pairwise CCR value same as that of Gold and large set Kasami sequences of same length. There are some set of segments of sequences with

73 21 pairwise CCR value less than that of Gold and large set Kasami sequences of same length and in both the cases linear complexity for most of the sequences is larger than that of Gold and large set Kasami sequences of same length. Also in the proposed scheme there is large number of choices for n apart from bifurcation parameter r and initial value x 0 in generating discrete and binary sequences. The smaller set of segments of binary sequences derived from chaotic sequences using mapping P 0, P 1, P 2, P 3 and binary sequences derived from chaotic sequence over GF(2 m ) with much less than Gold and large set Kasami sequences, satisfies the Welch bound and Sidelnikov bound better than Gold and large set Kasami sequences of same length. It is found that, BER is less using code sequences from the proposed set of segments of binary sequences whose peak magnitude of pairwise CCR value is less than that of Gold sequences and large set Kasami sequences. It is also seen that using the proposed set of segments of binary sequences derived from chaotic sequences, the BER is less based on the normalized mean square cross correlation value ( ) as compared to Gold sequences and large set Kasami sequences. Organization of other chapters in this thesis is given in next section. 1.9 ORGANIZATION OF THE THESIS Introduction to CDMA, Code sequences such as m-sequences, Gold sequences, Kasami sequences and chaotic sequences are presented in Chapter 1. Generation of chaotic discrete sequences and chaotic binary sequences using the proposed methods is also presented in this chapter. Chapter 2 provides a brief survey of the principles of generation of binary sequences derived from chaotic sequences for application in DS-CDMA.

74 22 In Chapter 3 Motivation and objectives of the present work are presented. In Chapter 4, mathematical preliminaries such as Groups, Rings and Fields are presented. Also this chapter discusses Trace Function, Hamming Cross Correlation function and Linear Complexity. Overview of properties of chaotic functions is also presented in this chapter. Chapter 5 presents the generation of chaotic discrete sequence over Z 4 and deriving binary sequence from this, using four methods. Chapter 6 presents the Generation of binary sequences from proposed Chaotic functions defined over Finite Field GF(2 m ). Three methods of transformation of sequence over GF(2 m ) to binary are presented. Chapter 7 presents the linear complexity and cross correlation properties of segments of sequences generated using the model proposed in Chapter 5 and Chapter 6. The results of such binary sequences of same length s, for s equal to 15, 31, 63, 127 and 255 generated using mapping P 0 and three Polynomial mappings P 1, P 2 and P 3 are presented in this chapter. The properties of chaotic binary sequences derived from chaotic functions defined over Finite Field GF(2 8 ) and GF(2 16 ) are investigated in Chpater 8. Chapter 9 presents bounds on CCR values of proposed sequences and Bit Error Rate (BER) performance of the proposed model. BER versus and BER versus number of users of the proposed sequences are compared with the Gold sequences and large set Kasami sequences. Chapter 10 concludes the thesis and suggests some directions for future work.

75 CHAPTER 2 LITERATURE SURVEY

76 23 CHAPTER 2 LITERATURE SURVEY 2.1 CHAOTIC FUNCTIONS AND SEQUENCES Chaos is a universal and robust phenomenon in many nonlinear systems (Tao Yang 2004). Although Poincare at the end of the nineteenth century, had noted that some mechanical systems could behave chaotically (Poincare H. 1892), chaos did not attract wide attention until Lorenz published his paper in 1963 (Lorenz E. N. 1963). When computers began to become available in the 1960s workers in fields such as meteorology and ecology were able to see the complex dynamics produced by straightforward equations, and this led to a burst of interest in both pure and applied mathematics that laid the foundations of the modern field of chaotic dynamics (Robert M. May and Angela R. McLean 2007). In population ecology, the classic paper is May s Simple mathematical models with very complicated dynamics (Robert M. May 1976). In 1976, Robert M. May was investigating the period doubling behavior of the logistic map, but did not understand what happened beyond the accumulation point of bifurcation parameter equal to Yorke and Tien-Yien Li (1975) had investigated the chaotic region of the map and coined the term Chaos. In 1980 s, Electrical Engineers first time officially announced the existence of chaos as noise-like behaviors in electronic circuits (Tao Yang 2004). Chaos is an aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions (Peter Stavroulakis 2006). Chaos describes the behavior of these dynamical systems that may exhibit non-periodicity and high dependence on initial conditions. The two types of dynamical systems are,

77 24 1. Discrete-time dynamical system that takes the current state as input and updates the situation by producing a new state as output. Such a system is usually modeled by a chaotic map. 2. Continuous-time dynamical system can be considered as the limit of discrete system with smaller and smaller updating time. Such a system is modeled by a differential equation. Since long time ago discrete-time chaotic systems had been used by the cryptographic community (Stinson D. 1995) to generate cipher keys, this fact led to the initiation of applying chaos to secure communication (Tao Yang 2004). The wideband and noise like features of a chaotic signal are particularly good in spread spectrum communications. There are several ways to spread a signal. The most practical methods of spread spectrum communications (Raymond L. Pickholtz et al 1982) are Direct Sequence Spread Spectrum (DSSS) and Frequency Hopping Spread Spectrum (FSSS). The use of chaos based system offered advantages over the conventional methods like large number of chaotic signals can be easily generated with an increase in system security hence provide advantage in terms of security, capacity and BER performance. Chaotic sequences in general are continuous amplitude discrete time non linear sequences having large period and two sequences generated with two initial values differing even by a small value are highly uncorrelated (Robert M. May 1976). But for digital communication application binary sequences are desirable. Hence, a study on the generation and properties of segments of binary sequences derived from chaotic sequences, suitable for applications in DS-CDMA is considered in this work. The chaotic sequences are generated by using different well known chaotic maps such as Logistic map, Tent map, Cubic map, Bernoulli map and Quadratic map (Stevan H. Strogatz 2000) and in this work methods are proposed to derive binary sequences from chaotic sequences. The properties of chaotic sequences generated using the six chaotic map equations are almost similar and discussed in Chapter 3.

78 EARLY METHODS OF DERIVING BINARY SEQUENCES FROM CHAOTIC SEQUENCES One of the most commonly used method of deriving binary sequences from chaotic sequences discussed in the literature is based on threshold function as reported in Parlitz U. and Ergezinger S. (1994), Tohru Kohda and Akio Tsuneda (1997), Hongtao Zhang et al (2000, 2001), Soobal Y etal (2002), Thiruvengadam S. J. et al (2003), Zouhair Ben Jemaa et al (2002, 2004, 2006), Youssef M. I. et al (2008 a, 2008 b) and Georges Kaddoum et al (2009). The other method of deriving binary sequences from chaotic sequences is using quatization and encoding of coupled map lattices (CML) as discussed in Hu Saigui (1996), Soumaya Meherzi et al (2006) and Ali Kotti et al (2010). Most of the sequences so generated fail to retain the properties of chaotic sequences. There are short cycle problems as reported in Julian Palmore et al (1990), Jessa M. (2002), Vladeanu C. et al (2003), Ding Qun et al (2005), Canyan Zhu (2008a, 2008b), Zheng Yan-bin and Ding Qun (2011) and Du Baoxiang et al (2013). This thesis focuses on deriving binary sequences from chaotic sequences having long period, good cross correlation and linear complexity properties. Applications of binary sequences derived using earlier methods such as using threshold function and using quatization and encoding of coupled map lattices, in DS- CDMA are discussed in the next section Applications of Chaotic Binary Sequences in DS-CDMA Once the idea of chaos based communication for secure transmission was pointed out, the researchers saw a potential advantage in using the chaotic sequences in DS- CDMA systems. As a result the concept of chaos based DS-CDMA was born. The application of chaotic sequences in DS-CDMA is discussed in the literature (Heidari-

79 26 Bateni G. and McGillem C. D. (1992) and (1994), Parlitz U. and Ergezinger S. (1994), Hu Saigui (1996), Tohru Kohda and Akio Tsuneda (1997a) and (1997b), Makoto Itoh (1999), Hongtao Zhang et al (2000) and (2001), Soobul Y. et al (2002), Zouhair Ben Jemaa et al (2002), (2004) and (2006), Thiruvengadam S. J. et al (2003), Soumaya Meherzi et al (2006), Youssef M. I. et al (2008 a) and (2008 b), Georges Kaddoum et al (2009), Ali Kotti et al (2010)). A brief comment on them is considered below. The application of chaotic sequences to a DS-CDMA system was first studied by Heidari-Bateni G. and McGillem C. D. (1992) and (1994). In this scheme, the nonperiodic chaotic non binary sequences generated using Logistic map replaces the binary spreading sequences used in DSSS systems. Instead of having only two values { 1, +1}, the amplitude of the spreading sequence now varies within a limited range. Such an arrangement makes the eavesdropper very difficult to estimate the spreading sequence being used. The BER performance of these non binary sequences in the presence of AWGN is compared with PN sequences of length 63 and 31. For an SNR of 10 db for 5 user system the BER is found to be approximately equal to for chaotic sequence of length 63 which is slightly less than that of PN sequences of same length (0.003). For an SNR of 10 db for 5 user system the BER is found to be approximately equal to for chaotic sequence of length 31 and for PN sequences of same length, the BER is found to be Under identical conditions, using proposed sequences it is found that the BER is less than and for sequences of length 31 and 63 respectively. A method of transmitting and encoding analog data by using chaotic carriers is discussed in Makoto Itoh (1999). The basic Direct Sequence technique is employed to encode and spread analog data sequences. The modulators consist of multiplication of chaotic carriers by the information signals. For 10 users with SNR 10 db, the probability of bit error is shown to be 0.09 and for sequences of length 100 and 200 respectively. Using proposed sequences, it is found that the BER is less and equal to for length 127 and for length 255 respectively.

80 27 The main advantages of chaos based SS techniques are dealing with both analog and binary information data (Makoto Itoh 1999). A chaotic signal is essentially analog and so chaos can deal with real numbers (analog data) directly. However the implementation is complex. Since most information is digital today, the chaos based DSSS techniques are mainly used to spread binary data sequences. Hence in the present day applications in DSSS, binary spreading sequences are preferred. The most commonly used method of deriving chaotic binary sequences is using one dimensional chaotic map by applying threshold function. Deriving binary sequences from chaotic sequences using threshold function is very simple with the mapping, x i > 0 mapped to 1 and x i < 0 mapped to -1. The properties of chaotic binary sequences generated using threshold function and their suitability in DSSS is discussed in (Parlitz U. and Ergezinger S. 1994), Tohru Kohda and Akio Tsuneda (1997), Hongtao Zhang et al (2000, 2001), Soobal Y etal (2002), Thiruvengadam S. J. et al (2003), Zouhair Ben Jemaa et al (2002, 2004, 2006), Youssef M. I. et al (2008 a, 2008 b) and Georges Kaddoum et al (2009)). Chaotic binary sequences generated using Logistic map equation and used as spreading sequences in spread spectrum technique is presented in (Parlitz U. and Ergezinger S. 1994). It is shown that this communication is reliable in the presence of Gaussian noise and several users may share the same bandlimited channel. For sequence of length 127 and in the presence of upto10 users, the bit error probability is shown to be less than for SNR of 8 db. It is found that under identical conditions, using the binary sequences proposed in this work, the BER is less than the above and is equal to Chaotic binary sequences derived from one dimensional Logisitc map and Chebyshev map and their performance in spread spectrum is discussed in Tohru Kohda and Akio Tsuneda (1997a). For 10 users the probability of bit error is shown to be equal to for length 31 and SNR 8 db. It is shown that using the proposed binary

81 28 sequences of same length, same number of users and same SNR, the BER is less than Based on Logistic map, Tent map, Bernoulli map and Chebyshev map, a simple method of deriving binary sequence is discussed in Hongtao Zhang et al (2000) and (2001). Such binary sequences are derived from chaotic sequences using threshold function. The maximum cross correlation value is less than for length 63. But using proposed sequences it is seen that there are segments of sequences of length 63 for less than Hence under identical conditions the BER is less. Generation of chaotic sequences using Logistic map and Tent map for application in CDMA is discussed in Soobul Y. et al (2002). Binary code sequences are derived from chaotic sequences using Logistic map and Tent map, using threshold function. The sequences of length 128 are shown to have maximum cross correlation value equal to 0.28 and probability of bit error equal to 0.1 for a 10 user system with SNR 8 db. Comparing this with the proposed sequences, it is found that for length 127, the maximum value of cross correlation is less than 0.134; BER is less than for 10 users. Chaotic binary sequences are generated using Logistic map and threshold function for an application to asynchronous DS-CDMA and are presented in Zouhair Ben Jemaa et al (2002), (2004) and (2006). The BER performance of these sequences are compared with Gold sequences and found to be better than that of Gold sequences. For sequences of length 31, for 10 users system with SNR 8 db the BER is Under identical conditions, the BER is less than using the proposed binary sequences. Generation of binary chaotic sequences using Tent map and threshold function is discussed in Thiruvengadam S. J. et al (2003). The loss of orthogonality between spreading waveforms in fading multipath channel is mitigated by employing multiuser

82 29 detection with multiple transmit and receive antennas. The performance of the system with space time multiuser detection is analyzed in fading Multipath channels with Additive White Gaussian Noise. It is shown that for 10 users with SNR 8 db, the BER is for chaotic binary sequences of length 31 and BER is for conventional PN sequences of same length. Comparing this with the proposed sequences it is found that the BER is less than for the same length with SNR 8 db and 10 users. The performance of chaotic code generators implemented in spread spectrum communication system is analyzed in Youssef M. I. et al (2008 a) and (2008 b). The chaotic sequences are generated using Logistic map and are transformed to binary using threshold function. It is shown that for the sequences of length 31, for 6 users with 8 db SNR the bit error probability is Under identical conditions, the BER is less than using the proposed sequences. An approach to compute the BER expression for multiuser asynchronous chaos based DS-CDMA system is presented in Georges Kaddoum et al (2009) and here a Piece-Wise Linear (PWL) map is chosen as a chaotic sequence generator (Charge P. et al 2008). The BER expression is computed in terms of the energy distribution, the number of paths, the noise variance and the number of users. It is reported that for 8 users with SNR 8 db bit error probability for spreading sequences of length 20, 40, 80, 160 and 320 is found to be 0.15, 0.07, 0.003, and respectively. The other method of deriving chaotic binary sequences is by using Coupled Map Lattice (CML) by applying quantization and encoding. The quantizer performs an equal interval quantization of the floating point number ranging from 0 to 1 or -1 to +1 and the coding block converts the quantized signal into a stream of bits. The properties of chaotic binary sequences and their suitability in DSSS reported in the literature using quantization and encoding of Coupled Map Lattice (CML) using Logistic map and is discussed in (Hu Saigui (1996) and (2004), Soumaya Meherzi et al (2006) and Ali Kotti et al (2010)). A brief note on them is as follows.

83 30 In paper Hu Saigui (1996), the peak value of cross correlation for chaotic binary sequences of length 127 derived from Logisitc map, is shown to be which is compared with Gold sequence of same length (peak CCR value equal to 0.13) and m- sequence of length 127 (peak CCR value equal to 0.32). Comparing this with the proposed sequences it is seen that there are set of sequences for peak CCR value less than for the same length. A family of spatiotemporal chaotic binary sequences derived from Logistic map and using quantization and encoding of CML is discussed in Soumaya Meherzi et al (2006), which as an alternative to the Gold codes used in asynchronous DS-CDMA systems. These new sequences are shown to have improved performance compared to the Gold codes. The performance criteria used in this study are Multiple Access Interference (MAI), Signal to Noise Ratio (SNR) and Bit Error Probability. It is shown that for a spreading factor of 31, for 10 users with SNR = 8 db, the BER is Under identical conditions, the proposed sequences have BER less than For a spreading factor of 31, the chaotic binary sequences derived using quantization and encoding of Coupled Map Lattice are compared with Gold sequences of same length and shown that probability of bit error is for Gold sequences and for CML for 10 simultaneous users with SNR 8 db in Ali Kotti et al (2010). It is found that using the proposed sequences of same length BER is less and equal to for 10 users and SNR 8 db. In the earlier published work of deriving binary sequences from chaotic sequences using threshold function, it is reported that the sequences have short cycle problems and is reported in Julian Palmore et al (1990), Jessa M. (2002), Vladeanu C. et al (2003), Ding Qun et al (2005), Canyan Zhu (2008a, 2008b), Zheng Yan-bin and Ding Qun (2011) and Du Baoxiang et al (2013).

84 31 In all the above research articles linear complexity and bound on peak value of cross correlation are not discussed. As mentioned in Chapter 1, a new family of segments of binary spreading sequences derived from chaotic sequences is proposed in this work and found to have following properties. 1) Linear Complexity (LC) is greater than that of Gold and large set Kasami sequences of same length, 2) The number of segments of sequences having same peak pairwise CCR value as that of Gold and large set Kasami sequences of same length is more than the number of Gold or large set Kasami sequences, 3) There are set of segments of sequences whose peak pairwise CCR value is less than that of Gold and large set Kasami sequences of same length, 4) A smaller set of Segments of binary sequences derived from chaotic sequences over Z 4 using mapping P 0, P 1, P 2, P 3 and binary sequences derived from chaotic sequence over GF(2 m ) having much lower than CCR values of Gold and large set Kasami sequences, satisfies the Welch bound and Sidelnikov bound better than Gold and large set Kasami sequences of same length and 5) Bit error probability is less by using proposed set of segments of sequences based on peak magnitude of pairwise CCR value and also on normalized mean square cross correlation ( ) value as compared to Gold sequences and large set Kasami sequences. Properties of the proposed segments of binary sequences such as period, LC and CCR properties are found to be better compared to the binary sequences obtained from

85 32 chaotic sequences using threshold function and quantization and encoding of coupled map lattice (CML) described in the literature survey. 2.3 SUMMARY Applications of binary sequences derived from chaotic sequences in DSSS Communication are discussed in the literature. In the literature, the two most commonly used methods of deriving the binary sequences from chaotic sequences is using threshold function and quantization and encoding of coupled map lattice (CML). Deriving binary sequences from chaotic sequences using threshold function is very simple way of generation of binary sequences from chaotic sequences with the mapping, x i > 0 mapped to 1 and x i < 0 mapped to -1. Because of non linear transfer characteristics such a scheme generates non linear binary sequence. Most of the sequences so generated fail to retain the properties of chaotic sequences such as long period and low value of cross correlation (Julian Palmore et al (1990), Jessa M. (2002), Vladeanu C. et al (2003), Ding Qun et al (2005), Canyan Zhu (2008a, 2008b), Zheng Yan-bin and Ding Qun (2011) and Du Baoxiang et al (2013)). The chaotic sequence over reals exhibits the properties of non linearity, non periodicity and sensitivity to initial conditions. Hence a method is proposed in this work to derive binary sequences from chaotic sequences which have similar properties as that of chaotic real valued sequences such as non linearity, large period and sensitivity to initial conditions. Generation of random binary sequences with better correlation properties is needed for spread spectrum communication system. DS-CDMA systems using chaotic binary sequences have been shown to posses certain advantages over conventional CDMA systems. Use of binary sequences derived from chaotic sequences are found to have better cross correlation properties than that of conventional sequences which improves the performance of CDMA systems.

86 33 Also the security of such CDMA systems is based on the Linear Complexity (LC) of the binary sequences. Higher LC ensures security of individual code sequence. If any user s code sequence is to be obtained by knowing a segment of sequence, a large segment is required, if LC is high. It is shown that, the proposed sequences have large LC and better CCR properties compared to conventional Gold sequences and large set Kasami sequences.

87 CHAPTER 3 OBJECTIVES OF THE PRESENT WORK

88 34 CHAPTER 3 OBJECTIVES OF THE PRESENT WORK 3.1 MOTIVATION It is found in the literature that most of the spread spectrum communication systems are based on the sequence generated using Linear Feedback Shift Registers (LFSR). Gold and Kasami Sequences are extensively used in CDMA based on Direct Sequence Spread Spectrum (DSSS) due to their excellent cross correlation properties, however as these codes are generated using LFSR s, their Linear Complexity is small. Other limiting factors are that they are relatively smaller in number and provide limited security (Mazzini G. et al 1997, 1998), as they can be identified with a number of samples which is much less than their actual length by means of linear regression models (Simon M. K. et al 1994). There has been extensive research on suitability of chaos based Direct Sequence Code Division Multiple Access (DS-CDMA) systems in recent years. It is found that, one of the most commonly used method of deriving binary sequences from chaotic sequences is based on threshold function as reported in Parlitz U. and Ergezinger S. (1994), Tohru Kohda and Akio Tsuneda (1997), Hongtao Zhang et al (2000, 2001), Soobal Y etal (2002), Thiruvengadam S. J. et al (2003), Zouhair Ben Jemaa et al (2002, 2004, 2006), Youssef M. I. et al (2008 a, 2008 b) and Georges Kaddoum et al (2009). The other method of deriving binary sequence from chaotic sequence is using quatization and encoding of coupled map lattices (CML) as discussed in Hu Saigui (1996), Soumaya Meherzi et al (2006) and Ali Kotti et al (2010). Most of the sequences so generated fail to retain the properties of chaotic sequences. There are short cycle problems as reported in Julian Palmore et al (1990), Jessa M. (2002),

89 35 Vladeanu C. et al (2003), Ding Qun et al (2005), Canyan Zhu (2008a, 2008b), Zheng Yan-bin and Ding Qun (2011) and Du Baoxiang et al (2013). Hence this thesis focuses on deriving binary sequences from chaotic sequences having long period, good cross correlation and linear complexity properties. 3.2 OBJECTIVES OF THE PRESENT WORK As discussed in Section 1.1.6, chaotic sequences are characterized by long period, non linearity and sensitivity to initial conditions which results in uncorrelated sequences. Objective of the work is to derive binary sequences from chaotic sequences which retain the desirable properties of chaotic sequences like non linearity, long period and low pairwise CCR values and are superior to the binary sequences obtained from the existing method of threshold function (Parlitz U. and Ergezinger S. (1994), Tohru Kohda and Akio Tsuneda (1997), Hongtao Zhang et al (2000, 2001), Soobal Y etal (2002), Thiruvengadam S. J. et al (2003), Zouhair Ben Jemaa et al (2002, 2004, 2006), Youssef M. I. et al (2008 a, 2008 b) and Georges Kaddoum et al (2009)) and quantization and encoding of coupled map lattice (Hu Saigui (1996) and (2004), Soumaya Meherzi et al (2006) and Ali Kotti et al (2010)). Methods of generating discrete sequences over Z 4 and GF(2 m ) and deriving binary sequences from them are proposed. Six chaotic functions discussed in Section are considered for generating chaotic discrete sequences over Z 4. As discussed in Section 1.4, four methods using mappings P 0, P 1, P 2 and P 3 are proposed to derive binary sequences from chaotic sequences over Z 4. Two chaotic map equations namely, Logistic map and a variation of Logistic map are considered for generating discrete sequences over GF(2 m ) for two standard values of m = 8 and m = 16. For deriving binary sequences from chaotic sequence over GF(2 m ) three methods are proposed as discussed in Section 1.5.

90 36 The segments of binary sequences of length 15, 31, 63, 127 and 255 so obtained are compared with Gold and large set Kasami sequences. The linear complexity of proposed sequences is investigated. The CCR values of set of sequences obtained are evaluated using correlation bounds. BER performance of proposed sequences in CDMA application is evaluated. In the subsequent chapters it is shown that the segments of binary sequences exhibit long period and better cross correlation properties than the earlier methods of obtaining binary sequences from chaotic sequences using threshold function and using quantization and encoding of CML. Mathematical preliminaries and overview of properties of chaotic functions are discussed in the next chapter.

91 CHAPTER 4 MATHEMATICAL PRILIMINARIES, OVERVIEW OF CHAOTIC FUNCTIONS AND PROPERTIES OF CHAOTIC SEQUENCES

92 37 CHAPTER 4 MATHEMATICAL PRILIMINARIES, OVERVIEW OF CHAOTIC FUNCTIONS AND PROPERTIES OF CHAOTIC SEQUENCES In this chapter, the mathematical preliminaries such as Groups, Rings, Fields, Polynomials over GF(p), Polynomial arithmetic, Irreducible polynomial, Primitive polynomial over GF(2), Binary Finite Fields, Trace Function, Hamming Cross Correlation function and Linear Complexity are discussed. Overview of properties of chaotic maps given by Equations (1.3) to (1.8) considered in this work, is also presented in this chapter. Mathematical preliminaries are considered in the next section. 4.1 MATHEMATICAL PRELIMINARIES Groups A set of elements {a, b, c } on which some binary operations are defined is called algebraic structure. A set of elements {a, b, c } with one binary operation denoted by * is said to be a group, if the following axioms are satisfied. i. Closure: for all a, b Є G, a * b is also in G ii. Associativity: for all a, b and c Є G, (a * b) * c = a * (b * c) iii. Identity: There is a unique element in G say e, such that for all a Є G, e*a=a*e = a iv. Inverse: for all a Є G, there exists a unique inverse of a denoted by a i such that a * a i = e

93 38 In addition to the above four axioms, if for all a, b Є G, a * b = b * a then the group is said to be Commutative group or Abelian group. The number of elements in a group is called order of the group. If the order of the group is finite then it is called as finite group. When the group operation is defined as addition, denoted by +, the group is called Additive group. For additive group, the identity element is denoted by 0 and additive inverse of any element say a is denoted by -a. When the group operation is defined as multiplication, denoted by x or. or no symbol, it is called multiplicative group. For multiplicative group the identity element is denoted by 1 and inverse of any element say a is denoted by a -1. The group operation + or x need not necessarily be usual arithmetic addition or multiplication. If there is an element say a in the group such that a * a, a * a * a etc. generates all the other elements in the group, then the group is called cyclic group generated by a. a itself is called generating element of the group Rings A set of elements R with two binary operations addition and multiplication defined and denoted by +, x is a Ring, if for all the elements a, b, c Є R the following axioms are satisfied. 1. R is an Abelian group under addition. The identity element is denoted by 0 and the additive inverse of a is denoted by -a.

94 39 2. under multiplication: i. Closure: for all a, b in R, a x b Є R ii. Associativity: for all a, b, c in R (a x b) x c = a x (b x c) iii. Distributivity: for all a, b, c in R, a x (b + c) = a x b + a x c (a + b) x c = a x c + b x c A ring is commutative if for all a, b Є R, a x b = b x a. A ring can be of finite elements or infinite elements, it may or may not have multiplicative identity. In a finite ring R there may be elements such that a 0, b 0 but a x b = 0. Such elements a, b are called zero divisors. An element which is not a zero divisor is called a unit element. Every unit element in the ring has a unique multiplicative inverse Fields Set F of elements {a, b, c } with two operations +, x such that, i. F is a commutative ring and ii. Every non-zero element has multiplicative inverse, then F is a field. Suppose the set of integers {0, 1, 2 (n 1)} with operations + and x modulo n is represented by Z n, then Z n in general is a commutative ring with identity. In Z 4 element, 2 is a zero divisor. Elements 1 and 3 are unit elements. Multiplicative inverse of 1 and 3 respectively are 1 and 3. Further,, for x = 1, 2, 3. Thus all elements in Z 4 can be expressed as positive integers.

95 40 If n = p, a prime, then Z p is a field of order p and is represented by GF(p) called, Galois field of order p. Order of any finite field is of the form p n, where p is a prime and n any positive integer. The finite field of smallest order is GF(2) = {0, 1} +, x modulo Polynomials Over GF(p) A polynomial of degree n over GF(p), (n 0) is an expression of the form, f(x) = a n x n + a n-1 x n-1 + a 1 x + a 0 (4.1) Where the coefficients a i Є GF(p) and a n 0. If n = 0, it is called zero th degree polynomial and if a n = 1, it is called monic polynomial Polynomial Arithmetic Let f(x) be a polynomial of degree n and g(x) be a polynomial of degree m over GF(p) with n m, a n 0 and b m 0, then addition, subtraction, multiplication and division can be defined as follows. Addition: (4.2) Subtraction: Where b i is additive inverse of b i in GF(p). (4.3)

96 41 Multiplication: (4.4) Where c k = (a 0 b k + a 1 b k-1 + a k-1 b 1 + a k b 0 ) mod p and a i = 0 for i > n and b i = 0 for i > m. Degree of the product polynomial is the sum of the degrees of individual polynomials. Division: Let f(x) be of degree n, g(x) be of degree m over GF(p) with n > m and suppose f(x) is divided by g(x). Let q(x) be quotient and r(x) be the remainder. Then we can write, f(x) = g(x) q(x) + r(x) (4.5) This can also be expressed as, f(x) modulo g(x) r(x) (4.6) Degree of r(x) is less than m Irreducible Polynomial A polynomial f(x) of degree n over GF(p) is said to be irreducible if it is not divisible by any polynomial other than 1 and itself Primitive Polynomial Over GF(2) Let f(x) be a polynomial of degree n over GF(2). The least integer k such that f(x) divides (x k - 1) is called order of the polynomial or exponent of the polynomial f(x). That is Let f(x) be an irreducible polynomial of degree n over GF(2). If its order k = (2 n 1), the polynomial f(x) over GF(2) is called primitive polynomial.

97 Binary Finite Fields Let f(x) be an irreducible polynomial over GF(2) of degree m, and let α be a root of f(x), i.e., f(α) = 0. Then, f(x) can be used to construct a binary finite field GF(2 m ) with exactly 2 m elements, where α itself is one of those elements. Furthermore, the set {1, α, α 2, α m-1 } forms a basis for GF(2 m ) and is called the polynomial basis of the field. Any arbitrary element b Є GF(2 m ) can be expressed in this basis as,, where coefficients are from GF(2) (4.7) All the elements in GF(2 m ) can be represented as polynomials of degree (m 1) or less. The order of an element λ Є GF(2 m ) is defined as the smallest positive integer k such that λ k = 1. In a polynomial basis representation of the elements of the binary finite fields, each element is represented as a binary string {b m-1 b 2 b 1 b 0 } which is equivalently considered a polynomial of degree less than m, (b m-1 x m b 2 x 2 + b 1 x 1 +b 0 ) (4.8) The addition of two elements a, b Є GF(2 m ) is simply the addition of two polynomials, where the coefficients are added in GF(2) or equivalently, the bitwise XOR operation on the coefficient vectors of polynomials corresponding to a and b. Multiplication is defined as the polynomial product of the two operands followed by a reduction modulo f(x). Finally, the inversion of an element α Є GF(2 m ) is the process to find an element α -1 Є GF(2 m ) such that α. α -1 1 mod f(x). Table 4.1 gives list of some irreducible polynomials over GF(2) (Solomon W. Goulomb and Guang Gong 2005).

98 43 Table 4.1 List of Irreducible Polynomials Over GF(2) Degree Irreducible Polynomial Over GF(2) n 1 x, (x + 1) 2 (x 2 + x + 1) 3 (x 3 + x + 1), (x 3 + x 2 + 1) 4 (x 4 + x + 1), (x 4 + x 3 + 1), (x 4 + x 3 + x 2 + x + 1) 5 (x 5 + x 2 + 1), (x 5 + x 3 + x 2 + x + 1), (x 5 + x 3 + 1), (x 5 + x 4 + x 3 + x + 1), (x 5 + x 4 + x 3 + x 2 + 1) 6 (x 6 + x 1 + 1), (x 6 + x 4 + x 2 + x 1 + 1), (x 6 + x 5 + x 2 + x 1 + 1), (x 6 + x 3 + 1), (x 6 + x 5 + x 3 + x 2 + 1), (x 6 + x 4 + x 3 + x 1 + 1), (x 6 + x 5 + x 4 + x 2 + 1), (x 6 + x 5 + x 4 + x 1 + 1), (x 6 + x 5 + 1) 7 (x 7 + x 1 + 1), (x 7 + x 5 + x 3 + x 1 + 1), (x 7 + x 3 + x 2 + x 1 + 1), (x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + 1), (x 7 + x 5 + x 4 + x 3 + 1), (x 7 + x 3 + 1), (x 7 + x 6 + x 5 + x 2 + 1), (x 7 + x 5 + x 4 + x 3 + x 2 + x 1 + 1), (x 7 + x 6 + x 5 + x 3 + x 2 + x 1 + 1), (x 7 + x 6 + x 3 + x 1 + 1), (x 7 + x 5 + x 2 + x 1 + 1), (x 7 + x 6 + x 5 + x 4 + x 2 + x 1 + 1), (x 7 + x 4 + 1), (x 7 + x 6 + x 4 + x 2 + 1), (x 7 + x 6 + x 4 + x 1 + 1), (x 7 + x 6 + x 5 + x 4 + 1), (x 7 + x 4 + x 3 + x 2 + 1), (x 7 + x 6 + 1) 8 (x 8 + x 4 + x 3 + x 2 + 1), (x 8 + x 6 + x 5 + x 4 + x 2 + x 1 + 1), (x 8 + x 7 + x 6 + x 5 + x 4 + x 1 + 1), (x 8 + x 6 + x 5 + x 3 + 1), (x 8 + x 7 + x 5 + x 4 + x 3 + x 2 + 1), (x 8 + x 7 + x 6 + x 5 + x 2 + x 1 + 1), (x 8 + x 5 + x 3 + x 1 + 1), (x 8 + x 7 + x 6 + x 4 + x 2 + x 1 + 1), (x 8 + x 6 + x 5 + x 2 + 1), (x 8 + x 7 + x 3 + x 1 + 1), (x 8 + x 6 + x 5 + x 1 + 1), (x 8 + x 4 + x 3 + x 1 + 1), (x 8 + x 5 + x 4 + x 3 + x 2 + x 1 + 1), (x 8 + x 7 + x 3 + x 2 + 1), (x 8 + x 5 + x 3 + x 2 + 1), (x 8 + x 6 + x 4 + x 3 + x 2 + x 1 + 1), (x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + 1), (x 8 +x 7 + x 6 + x 1 + 1), (x 8 + x 5 + x 4 + x 3 + 1), (x 8 + x 7 + x 5 + x 3 + 1), (x 8 + x 7 + x 2 + x 1 + 1), (x 8 + x 7 + x 5 + x 4 + 1), (x 8 + x 6 + x 3 + x 2 + 1), (x 8 + x 7 + x 6 + x 3 + x 2 + x 1 + 1), (x 8 + x 7 + x 6 + x 4 + x 3 + x 2 + 1), (x 8 + x 7 + x 5 + x 1 + 1), (x 8 + x 7 + x 6 + x 5 + x 4 + x 2 + 1), (x 8 + x 7 + x 4 + x 3 + x 2 + x 1 + 1), (x 8 + x 6 + x 5 + x 4 + x 3 + x 1 +1), (x 8 +x 6 +x 5 +x 4 + 1) 16 (x 16 + x 5 + x 3 + x 2 + 1), (x 16 + x 8 + x 6 + x 5 + x 4 + x 1 + 1), (x 16 + x 11 + x 10 + x 9 + x 8 + x 7 + x 5 + x 4 + x 3 + x 2 + 1), (x 16 + x 14 + x 12 + x 11 + x 9 + x 6 + x 4 + x 2 + 1), (x 16 + x 12 + x 11 + x 10 + x 9 + x 8 + x 5 + x 3 + x 2 + x 1 + 1), (x 16 + x 15 + x 14 + x 13 + x 12 + x 10 + x 9 + x 8 + x 7 + x 5 + x 4 + x 2 + 1), (x 16 + x 15 + x 13 + x 11 + x 10 + x 9 + x 8 + x 6 + x 5 + x 2 + 1), (x 16 + x 14 + x 13 + x 12 + x 9 + x 8 + x 6 + x 5 + x 2 + x 1 + 1), (x 16 + x 11 + x 10 + x 9 + x 7 + x 5 + 1), (x 16 + x 5 + x 3 + x 2 + 1) etc Trace Function Definition: The trace function, denoted by Tr(β) of an element β Є GF(2 m ), relative to GF(2) is defined by, = (4.9)

99 44 The trace of an element of GF(2 m ) relative to GF(2) is a binary number, either 0 or 1. The trace function is a linear mapping from GF(2 m ) to GF(2). In this work two finite fields GF(2 8 ) and GF(2 16 ) are considered. Properties of Trace Function (Solomon W. Coulomb and Guang Gong 2005): i. for all α, β Є GF(2 m ) ii. for i = 1, 2, 3 (m - 1) iii. For O Є GF(2 m ), Є GF(2) Hamming Cross Correlation (HCCR) of Sequences Definition: The normalized cyclic Hamming cross correlation function (Abraham Lempel and Haim Greenberger 1974) of two sequences x and y of length N symbols is defined as (4.10) where 0 N-1 and is equal to the number of locations by which one sequence say y is cyclically shifted with respect to the other sequence x. n(τ) and d(τ) are the number of locations at which symbols agree and disagree respectively between the two sequences x and y Linear Complexity of Sequences Definition: Linear complexity (Massey J.L. 1969) of a binary sequence of finite length is the length of the shortest LFSR that generates the same sequence. Berlekamp Massey algorithm is an efficient algorithm for determining the linear complexity of binary sequence of finite length. The binary levels 1, 0 instead of 1, -1 are used to apply Berlekamp Massey algorithm.

100 45 Suppose a sequence of length N is of low linear complexity, then with the knowledge of a small segment of sequence it is possible to construct the entire sequence. On the other hand if LC is high, unless a large segment of sequence is known it is difficult to construct the sequence. This will provide inherent security in maintaining the secrecy of the sequence when used as key sequence in cryptographic and other applications. The properties of Logistic map equations are considered in the next section, followed by properties of other map equations. 4.2 LOGISTIC MAP EQUATIONS The Logistic Map given by Equation (1.3) which is repeated here, where 0 < x < 1 (4.11) The Logistic map was studied in 1976 by Robert M. May. An autonomous 1-D prototypical example from the discrete dynamical system class known as the logistic map used to model population dynamics is discussed in Robert M. May (1976). Depending on the value of bifurcation parameter r, the dynamics of this system can change dramatically, exhibiting periodicity or chaos. For 3.57 < r < 4, the sequence is found to be non periodic and non-converging. Control parameter r is in the range, 0 < r < 4 so that Logistic map given by Equation (4.11) maps in the interval 0 x 1. In Equation (4.11), for r = 0.75, r = 2.00 and r = 2.75 with initial value x 0 = 0.5, the trajectories for 100 iterations are illustrated in Figure 4.1. In Figure 4.1, if r = 0.75 (i.e., if r < 1) then, as. For 1 < r < 3, then trajectories reach a non-zero steady state and after few number of iterations the sequence converges to a single value. Such a system which settles down to a single value is called as one-point attractor or period-1 cycle.

101 46 Figure 4.1 One-Point Attractor of Logistic Map given by Equation (4.11). When r = 3.2 (i.e., if r > 3) the system settles down between two points i.e., x k repeats for every two iterations. This is called a two-point attractor or Period - 2 cycle. When r = 3.54 the system settles down between four points and hence called a four-point attractor or Period - 4 cycle. When r = 3.99, results in an N-point attractor as shown in Figure 4.2. Period doublings to cycles 2, 4, 8, 16, 32 and so on occurs as r increases. r is chosen to be in the range 3.57 < r < 4. Figure 4.2 N-point Attractor of Logistic Map given by Equation (4.11).

102 47 Figure 4.3 shows bifurcation diagram of the Logistic map given by Equation (4.11). A Bifurcation diagram is a visual summary of the succession of period doubling produced as r increases in the Equation (4.11). For each value of r, the system is first allowed to settle down and then the successive values of x are plotted for few hundred iterations. Figure 4.3 Bifurcation Diagram of Logistic Map given by Equation (4.11). The sensitivity to initial values of Equation (4.11) is shown in Figure 4.4. As illustrated in Figure 4.4, with r = 3.99 and using slightly different initial values x 0 = 0.3 and x 0 = , the trajectories follow two separate orbits after about 20 iterations. Hence even a small change in the initial value, results in two different uncorrelated sequences after few iterations.

103 48 Figure 4.4 Sensitivity to Initial Conditions of Logistic Map given by Equation (4.11). Next consider Logistic map given by Equation (1.4) which is repeated here, where -1 < x < 1 (4.12) Here for 1.72 < r < 2, the sequence is found to be non-periodic and nonconverging. If r = 0.8, the system settles down to two points and is called 2 point attractor. If r = 1.25 the system settles down to four points and is called a 4 point attractor. If r = 1.39 the system settles down to eight points and is called an 8 point attractor. If r = 1.99, it is called an N point attractor. Sensitivity to initial conditions is shown in Figure 4.5, for r = 1.99, initial value x 0 = 0.3 and for 50 iterations. Here also the trajectories follow two different orbits after about 22 iterations.

104 49 Figure 4.5 Sensitivity to Initial Condition of Logistic Map given by Equation (4.12). 4.3 TENT MAP EQUATION The Tent map is given by Equation (1.5) which is repeated here, where 0 < x < 1 (4.13) The sensitivity to initial condition is shown in Figure 4.6, for initial values 0.3 and for 50 iterations.

105 50 Figure 4.6 Sensitivity to Initial Condition of Tent Map given by Equation (4.13). 4.4 CUBIC MAP EQUATION The Cubic map is given by Equation (1.6) which is repeated here, where -1 < x < 1 (4.14) The sensitivity to initial condition can be seen for initial values 0.3 and for 50 iterations and is shown in Figure 4.7.

106 51 Figure 4.7 Sensitivity to Initial Condition of Cubic Map given by Equation (4.14). 4.5 QUADRATIC MAP EQUATION The Quadratic map is given by Equation (1.7) which is repeated here, where < x < 0.5 (4.15) For 0.36 < r < 0.5, the sequences are found to be non-periodic and nonconverging. The sensitivity to initial condition can be seen for initial values 0.3 and for 50 iterations and is shown in Figure 4.8 and Bifurcation diagram is shown in Figure 4.9.

107 52 Figure 4.8 Sensitivity to Initial Condition of Quadratic Map given by Equation (4.15). Figure 4.9 Bifurcation Diagram of Quadratic Map given by Equation (4.15).

108 BERNOULLI MAP EQUATION The Bernoulli map is given by Equation (1.8) which is repeated here, where < x < 0.5 (4.16) For 1.2 < r < 2, the sequences are found to be non-periodic and non-converging. If r = then the system settles down between two points and is called a 2 point attractor. The sensitivity for initial value 0.3 and for 50 iterations is shown in Figure 4.10 and the Bifurcation diagram is shown in Figure Figure 4.10 Sensitivity to Initial Condition of Bernoulli Map given by Equation (4.16).

109 54 Figure 4.11 Bifurcation Diagram of Bernoulli Map given by Equation (4.16). All the above chaotic equations can be regarded as first order non linear recurrence relation (NLRR) over reals. As mentioned earlier, for proper choice of bifurcation parameter r, the chaotic sequences are found to have long period. They are non linear and have sensitivity to initial values. Generation of chaotic discrete sequences over Z 4 and chaotic binary sequences derived from them is discussed in the next chapter. 4.7 SUMMARY In this chapter mathematical preliminaries related to Groups, Rings, Fields, Polynomials over GF(p), Polynomial arithmetic, Irreducible polynomial, Primitive polynomial over GF(2), Binary Finite Fields, Trace function, Hamming Cross Correlation and Linear Complexity are discussed.

110 55 The properties of all the six chaotic map equations which can be considered as first order non linear recurrence relation (NLRR) are also studied. In all the six chaotic map equations it is found that the real valued sequences exhibit chaotic behavior for proper choice of bifurcation parameter and the initial value within the range. Also they exhibit sensitivity to initial conditions. The inherent properties of these chaotic functions are used to generate proposed discrete and binary sequences. The properties of the proposed sequences are studied in Chapter 7.

111 CHAPTER 5 GENERATION OF CHAOTIC DISCRETE AND BINARY SEQUENCES

112 56 CHAPTER 5 GENERATION OF CHAOTIC DISCRETE AND BINARY SEQUENCES Deriving chaotic binary sequences from chaotic functions using threshold function are defined in the literature (Parlitz U. and Ergezinger S. 1994), Tohru Kohda and Akio Tsuneda (1997a) and (1997b), Hongtao Zhang et al (2000, 2001), Soobal Y etal (2002), Thiruvengadam S. J. et al (2003), Zouhair Ben Jemaa et al (2002, 2004, 2006), Youssef M. I. et al (2008 a, 2008 b) and Georges Kaddoum et al (2009)). Although the chaotic binary sequence generation process is simple using threshold function, most of the sequences fail to retain the properties of chaotic sequences (Julian Palmore et al (1990), Jessa M. (2002), Vladeanu C. et al (2003), Ding Qun et al (2005), Canyan Zhu (2008a, 2008b), Zheng Yan-bin and Ding Qun (2011) and Du Baoxiang et al (2013)). The main aim of the proposed work is to derive binary sequences from chaotic sequences which retain the desirable properties such as long period, non-linearity and low pairwise CCR values better than the earlier methods proposed in the literature. In this chapter a model to generate chaotic discrete sequences and segments of binary sequences are derived from discrete sequences using six chaotic map equations such as Logistic map, Tent Map, Cubic Map, Quadratic Map and Bernoulli Map is proposed. Proposed scheme obtains binary sequences from chaotic sequence in two steps. A discrete sequence over Z 4 is generated choosing appropriate multiplication factor n which multiplies each element of the sequence. The integer part is reduced to

113 57 modulo 4. The sequence generated has got 4 levels 0, 1, 2 and 3. Binary sequence is derived from sequence over Z 4 by Mapping P 0 based on binary conversion of integers 0, 1, 2 and 3. Polynomial mapping P 1, P 2 and P 3 (Udaya P and Siddiqi M.U. 1996) from Z 4 to binary. It is shown that segments of binary sequences so generated are non-linear with long period. The details are discussed in the following section. 5.1 GENERATION OF DISCRETE CHAOTIC SEQUENCES OVER Z 4 For any chaotic function, by proper choice of bifurcation parameter r as specified, any initial value x 0, iteratively generates a real valued infinite sequence {x i }, i = 0, 1, 2..., where with x 0 in the specified range depending on the chaotic function, makes each x k to lie in the same range. Each element x k of sequence {x i } is then multiplied by an integer n, n > 4 and the fraction part is discarded as explained in Section 1.3. The integer part is Q k. The integer part Q k is then reduced to small integer y k modulo 4. It is necessary that n > 4. The proposed model to generate discrete sequences from chaotic sequence {x k } is shown in Figure 5.1. The scheme is governed by equation where n > 4 (5.1) Where integer part of product x k n. As mentioned earlier, the resulting discrete sequence {y k } has 4 finite levels 0 y k < 4 and hence is over Z 4. Since n should be greater than 4, in this work six values for n that is 5, 6, 7, 8, 9 and 10 are considered. With these values for n, different discrete sequences {y k } are generated for different initial conditions. It is shown in Chapter 7 that for choice of different multiplication factor n = 5, 6, 7, 8, 9 and 10, it is possible to generate discrete sequences exhibiting good normalized cyclic Hamming cross correlation and balance properties.

114 58 x k ; k = 1, 2, 3,.. x 0 (Initial value) Chaotic Function x k+1 Integer Part Q k Mod 4 y k D n Figure 5.1 Scheme of Generating Chaotic Discrete Sequence. Methods of deriving binary sequences from discrete chaotic sequences over Z 4 are considered in the next section. 5.2 GENERATION OF CHAOTIC BINARY SEQUENCES Elements of discrete sequence {y k } over Z 4 so generated as explained in Section 5.1 are mapped to binary to get binary sequence {b k } using the mappings as discussed in Section 1.4 and is repeated here. Mapping P 0 based on binary conversion is considered first Mapping P 0 Based on Binary Conversion Let y Є Z 4, then P 0 (y) = b 1 b 0, where b 1, b 0 Є {0,1}, such that b b = y k. The mapping P 0 is essentially representation of elements of Z 4 namely 0, 1, 2 and 3 in binary P 0 (0)=00, P 0 (1)=01, P 0 (2)=10 and P 0 (3)= Polynomial Mapping P 1, P 2 and P 3, from Z 4 to Binary (Udaya P and Siddiqi M.U., 1996) a) P 1 (y) = 2y mod 4 (5.2) P 1 : y Є Z 4 is mapped to {0, 2} and {0, 2} is further mapped to (1, -1) P 1 can also be regarded as a mapping y Є Z 4 mapped to and then -1 is regarded as 0.

115 59 b) P 2 (y) = (y 2 y) mod 4 (5.3) P 2 : y Є Z 4 is mapped to {0, 2} and {0, 2} is further mapped to (1, -1) P 2 can also be regarded as a mapping y Є Z 4 mapped to and -1 is regarded as 0. c) P 3 (y) = (y 2 + y) mod 4 (5.4) P 3 : y Є Z 4 is mapped to {0, 2} and {0, 2} is further mapped to (1, -1) P 3 can also be regarded as a mapping y Є Z 4 mapped to and -1 is regarded as 0. The element y Є Z 4 and corresponding mapped element is given in Table 5.1. Table 5.1 Polynomial Mappings P i (y), i = 1, 2, 3. y Є Z P 1 (y) = 2y mod P 2 (y) = (y 2 y) Pmod4 3 (y) = (y 2 + y) mod Thus the model shown in Figure 5.1 can be used to derive discrete sequence {y k } over Z 4 ; where y k Є {0, 1, 2, 3} or binary {b k }, from the chaotic sequence {x k }. The linear complexity and CCR properties of the binary sequences derived from chaotic sequence over Z 4 are discussed in Chapter 7. Having discussed generation of discrete sequence over Z 4 and binary sequence using mapping P 0 and using three polynomial mappings P 1, P 2 and P 3, in the next chapter, chaotic sequences over GF(2 m ) and binary mappings are presented. 5.3 SUMMARY In this chapter, a scheme for generation of discrete sequences over Z 4 using chaotic map equations is proposed. The generated discrete sequences are transformed to binary sequences using mappings P 0, P 1, P 2 and P 3.

116 CHAPTER 6 GENERATION OF BINARY SEQUENCES FROM PROPOSED CHAOTIC FUNCTIONS DEFINED OVER FINITE FIELD GF(2 m )

117 60 CHAPTER 6 GENERATION OF BINARY SEQUENCES FROM PROPOSED CHAOTIC FUNCTIONS DEFINED OVER FINITE FIELD GF(2 m ) In this chapter chaotic functions over GF(2 m ) are defined and generation of discrete sequences using chaotic functions defined over finite field GF(2 m ) is proposed. The results indicate that for appropriate choice of bifurcation parameters and initial values both from GF(2 m ), a periodic sequence of period (2 m 1) can be obtained. These sequences over GF(2 m ) are transformed to binary using three techniques, as discussed in Section 1.5, i) Expressing every element in GF(2 m ) of the sequence as binary m tuple ii) Selecting a particular binary bit from each element of the sequence over GF(2 m ) iii) Mapping every element in GF(2 m ) to GF(2) using Trace function In this work two finite fields corresponding to two standard word sizes, m = 8 and m = 16 are considered. It is seen from computer search that all the six chaotic maps do not generate discrete sequences over GF(2 m ) for m = 8 or 16 with appreciable period. Only Logistic map and its variation generates sequences over GF(2 m ) for m = 8 or 16 with appreciable period. Hence in this study only Logistic map and its variation are considered.

118 DESCRIBING CHAOTIC FUNCTIONS OVER FINITE FIELD GF(2 m ) The concept of chaotic map equations over reals as discussed in Chapter 5, is extended to chaotic map equation defined over finite field GF(2 m ) in this chapter. Proposed chaotic map equations over GF(2 m ) are based on Logistic map and is given by, (6.1) where x 0, r 1 and r 2 are all from GF(2 m ). Theoretically, the Equation (6.1) can be defined for m 1. In this work m = 8 and m = 16 are considered as 8 and 16 are standard word sizes. As discussed in Section 1.5, the chaotic map equation (6.1) can be regarded as a first order non linear recurrence relation (NLRR). The properties of the sequences depend on the choice of initial value x 0. For certain values of x 0 the sequence generated is trivial. This is depicted in Note 6.1. Note 6.1: Let the chaotic map be given by equation over GF(2 m ). i) If x 0 = 0 or r 1 = 0, then the resulting sequence is a trivial sequence of all zeros. ii) If x 0 = r 2, then the sequence will be r 2, 0, 0, 0, 0, Hence in Equation (6.1) these cases are avoided. Thus for a given choice of r 1 0, any r 2 and if x 0 r 2, sequence generated is not trivial all zero sequence. A variation of Logistic map is defined by, (6.2) where r 1, r 2 and x k Є GF(2 m ). As discussed in Section 1.5, Equation (6.2) can be regarded as a first order inhomogeneous linear recurrence relation over GF(2 m ). Theoretically, the Equation (6.2) can be defined for m 1. However, investigation is carried out for two

119 62 standard word lengths m = 8 and m =16.When Equation (6.2) is used some sequences are trivial depending on the choice of initial values as given in Note 6.2. Note 6.2: Let the chaotic map be given by Equation (6.2) over GF(2 m ). If x 0 = r 2 (r ) -1, then the sequence generated is a trivial sequence of x 0. Proof: Consider equation,. If for k = 0 or x 0 = r 2 (r ) -1 Then x 1 = x 0, x 2 = x 0 and hence the elements repeat resulting in a trivial sequence of x 0. Note: For r 1 0 the sequence generated is not an all zero sequence and for x 0 r 2 (r ) -1, the sequence generated is not an all x 0 sequence. section. Generation of chaotic sequence over finite field GF(2 m ) is considered in the next 6.2 GENERATION OF CHAOTIC SEQUENCE OVER FINITE FIELD GF(2 m ) The block diagram for generating chaotic sequence over GF(2 m ) is as shown in Figure 6.1. The chaotic sequence generator shown in the block diagram generates chaotic

120 63 sequence of field elements defined over GF(2 m ) for different initial values x 0 and the bifurcation parameters r 1 & r 2. x k ; k = 1, 2, 3, x 0 (Initial value) Chaotic Function over GF(2 m ) x k+1 Binary Mapping Binary Sequence {b k } D Figure 6.1 Block Diagram for Generation of Chaotic Sequence Over GF(2 m ). The random chaotic sequence over GF(2 m ) generated using Equations (6.1) and (6.2) are mapped to binary using the three different techniques as discussed in Section 1.5. In this work, finite fields GF(2 8 ) and GF(2 16 ) are considered. Generation of chaotic sequence over finite field GF(2 8 ) and mapping to binary is considered in the next section. 6.3 CHAOTIC SEQUENCE OVER FINITE FIELD GF(2 8 ) Finite field GF(2 8 ) with addition multiplication modulo a primitive polynomial (x 8 +x 6 +x 5 +x 4 +1) over GF(2) is considered. Let g be a root of equation (x 8 +x 6 +x 5 +x 4 +1) = 0. g is of order 255 in GF(2 8 ). That is g 255 = 1. Then g is a primitive or generating element whose powers g 2, g 3 g 255 along with g give all the nonzero elements in GF(2 8 ). Further every nonzero element of the form g j for j 8 can be expressed uniquely as a polynomial of degree less than or equal to 7 over GF(2). Every polynomial of degree 7 or less is of the form (b 7 x 7 + b 6 x 6 + b 1 x + b 0 ), where b i Є (0, 1). Such polynomials can uniquely be represented in terms of its coefficients b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0. Thus every nonzero element in GF(2 8 ) can be represented by g j for some j, 1 j

121 or by a corresponding binary 8 tuple. The 0 in GF(2 8 ) is denoted by 8 tuple of all zeros. The sequences generated is a random sequence of field elements from GF(2 8 ). The computation over GF(2 8 ) is polynomial addition multiplication modulo (x 8 +x 6 +x 5 +x 4 +1) over GF(2) and is carried out using computer simulation. r 1, r 2 and x k can take the values 0, g, g 2, g 3 g 255. By selecting proper initial values x 0 and bifurcation parameters r 1, r 2 in Equation (6.1) and Equation (6.2), sequences over GF(2 8 ) of maximum possible period is generated. Given x k, x k+1 is computed using arithmetic modulo primitive polynomial (x 8 +x 6 +x 5 +x 4 +1) over GF(2). This arithmetic in GF(2 8 ) is implemented in the computer using Matlab program. For arbitrarily chosen values of x 0, r 1 and r 2, the maximum possible period of the sequence over GF(2 8 ) is determined by computer search and found to be 63 for chaotic mapping defined by Equation (6.1) and 255 for chaotic mapping defined by Equation (6.2). The random sequence of finite field elements generated using chaotic map Equations (6.1) and (6.2) over GF(2 8 ) as explained above are mapped to binary using the three techniques and is presented in Sections 6.3.1, and Expressing Every Element in GF(2 8 ) of the Sequence as Binary 8 Tuple Each of the finite field elements are expressed as binary 8 tuple and concatenated to get binary sequence. Hence the period of the binary sequence is 63 x 8 = 504 using Equation (6.1) and 255 x 8 = 2040 using Equation (6.2).

122 Selecting a Particular Bit from Each Element of the Sequence Over GF(2 8 ) In this method of binary mapping, for arbitrarily chosen values of x 0, r 1 and r 2, the maximum possible period sequence over GF(2 8 ) is generated. Each of the finite field elements are expressed as binary 8 tuple. First bit (MSB) of all the resulting binary 8 tuple are selected and concatenated to get a binary sequence of maximum possible period. In the same way 2 nd bit to 8 th bit (LSB) are selected and corresponding bits are concatenated. Hence choosing bits in location i (i = 1 to 8) results in 8 sequences of maximum possible period. The maximum period of the binary sequence by choosing bits in location i (i = 1 to 8) is 63 for each i, in case of Equation (6.1) and 255 for each i in case of Equation (6.2). Hence in case of binary sequences generated by choosing i th bit, the maximum possible period of the binary sequence is same as that of sequence over GF(2 8 ) Mapping Every Field Element in GF(2 8 ) to GF(2) Using Trace Function The trace function denoted by Tr(β) of any element β Є GF(2 8 ), relative to GF(2) is given by the equation, (6.3) The arithmetic is modulo primitive polynomial (x 8 +x 6 +x 5 +x 4 +1) over GF(2). By definition, for O Є GF(2 8 ), Tr(O) = 0 Є GF(2). Also for β Є GF(2 8 ), for i = 1, 2, 3 7

123 66 Other properties of Trace function is given in earlier Section Table 6.1 gives mapping using trace function for some elements in GF(2 8 ). The sequence of elements from GF(2 8 ) generated using the chaotic maps as discussed in the previous section, are expressed as powers of g or polynomials in g of degree 7 or less over GF(2) and are mapped to GF(2) using Trace function to one bit binary (either 0 or 1) and concatenated. Table 6.1 Some Mapping using Trace Function for Elements in GF(2 8 ) Element in GF(2 8 ) Polynomial b 7 g 7 + b 6 g b 1 g 1 + b 0 Binary 8 tuple b 7 b 6 b 1 b 0 Trace (.) Є GF(2) g b 1 g g 2 b 2 g g 3 b 3 g g 7 b 7 g g 128 b 7 g 7 + b 6 g b 1 g 1 + b In this case also, the maximum possible period of the binary sequence is same as that of sequence over GF(2 8 ). As mentioned earlier the maximum possible period of sequence over GF(2 8 ) is determined by computer search and found to be 63 in case of Equation (6.1) and 255 in case of Equation (6.2). Example 6.1 illustrates generation of segment of sequence of five elements over GF(2 8 ) and corresponding binary sequences using Equation (6.1). The field arithmetic is implemented in MATLAB. Example 6.1: Consider Equation (6.1) defined over GF(2 8 ). Suppose x 0 = g, r 1 = g 23 and r 2 = g 24. Put x 0 = g, r 1 = g 23 and r 2 = g 24 in Equation (6.1) to get,

124 67, since. The first five elements of the above chaotic sequence with initial value g is g, g 107, g 180, g 214, g 99 Sequence elements over GF(2 8 ), corresponding binary 8 tuple, binary bits in location i, for i = 1, 2, 3 8 and trace function values are listed in Table 6.2. Values of Field element Table 6.2 Sequence Generated in Example (6.1) Expressing elements as binary 8 tuple Bits in location i i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 Trace function x 0 g x 1 g x 2 g x 3 g x 4 g Binary sequence obtained by expressing every element as 8 tuple which are concatenated is given by, Segments of 8 binary sequences obtained by selecting i th bit are, for i = 1: as given by column 4 i = 2: as given by column 5 i = 3: as given by column 6 Similarly for i = 8: as given by column 11

125 68 Binary sequence obtained using trace function is, as given by last column Example 6.2 illustartes generation of sequences over GF(2 8 ) and corresponding binary sequences using Equation (6.2). Example 6.2: Consider Equation (6.2) over GF(2 8 ). Suppose x 0 = g, r 1 = g and r 2 = g. Put x 0 = g, r 2 = g in Equation (6.2) to get, = 0 = g 2 The first five elements of the chaotic sequence over GF(2 8 ) is g, 0, g 2, g 233, g 61 Sequence elements over GF(2 8 ), corresponding binary 8 tuple, binary bits in location i, for i = 1, 2, 3 8 and trace function values are listed in Table 6.3. Values of Field element Table 6.3 Sequence Generated in Example (6.2) Expressing elements as binary 8 tuple Bits in location i i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 Trace function x 0 g x x 2 g x 3 g x 4 g The binary sequence obtained by expressing every element as 8 tuple which are concatenated is given by,

126 69 Segments of 8 binary sequences obtained by selecting i th bit are, for i = 1: as given by column 4 i = 2: as given by column 5 i = 3: as given by column 6 Similarly for i = 8: as given by column 11 Binary sequence obtained using trace function is, as given by last column Chaotic function over GF(2 16 ), generation of discrete chaotic sequence over GF(2 16 ) and deriving binary sequence from this are discussed next. 6.4 CHAOTIC SEQUENCE OVER FINITE FIELD GF(2 16 ) Finite field GF(2 16 ) with addition multiplication modulo a primitive polynomial (x 16 +x 5 +x 3 +x 2 +1) over GF(2) is considered. There are 2 16 elements in GF(2 16 ). Let g be a root of equation x 16 +x 5 +x 3 +x 2 +1 = 0. Then g is a primitive or generating element whose powers g 2, g 3 g along with g give all the nonzero elements in GF(2 16 ). Further every nonzero element of the form g j for j 16 can be expressed uniquely as a polynomial of degree less than or equal to 15 over GF(2). Every polynomial of degree 15 or less is of the form b 15 x 15 + b 14 x b 1 x + b 0, where b i Є (0, 1). Such polynomials can uniquely be represented in terms of its coefficients b 15 b 14 b 13 b 4 b 3 b 2 b 1 b 0. Thus every nonzero element in GF(2 16 ) can be represented by g j for some j, 1 j or by a polynomial of degree less than 16 or by a corresponding binary 16 tuple. r 1, r 2 and x k can take the values 0, g, g 2, g 3 g By selecting proper initial values x 0 and bifurcation parameters r 1, r 2 in Equation (6.1) and Equation (6.2), sequences over GF(2 16 ) with maximum possible period are generated. Only those nontrivial values of x 0, r 1 and r 2 are used in Equation (6.1) and (6.2) which generate sequences with maximum possible period. Given x k, x k+1 is computed using arithmetic modulo primitive polynomial (x 16 +x 5 +x 3 +x 2 +1) over GF(2). This arithmetic in GF(2 16 ) is

127 70 implemented in the computer using Matlab program. For arbitrarily chosen values of x 0, r 1 and r 2 in Equation (6.1), the maximum possible period of the sequence over GF(2 16 ) is determined by computer search and found to be The maximum possible period of sequence over GF(2 16 ) is = using Equation (6.2). Sequences with maximum period are considered. The random sequence of finite field elements generated using chaotic map Equations (6.1) and (6.2) over GF(2 16 ) as explained above are mapped to binary using three different methods and is presented in Section to Expressing Every Element in GF(2 16 ) of the Sequence as Binary 16 Tuple Each of the finite field elements are expressed as binary 16 tuple and concatenated to get binary sequence. Hence the period of the binary sequence is x 16 = using Equation (6.1) and x 16 = using Equation (6.2) Selecting a Particular Bit from Each Element of the Sequence Over GF(2 16 ) Binary sequence is obtained from sequence over GF(2 16 ) by choosing bits in location i (where i = 1 (MSB) to 16 (LSB)) of each element in the sequence. The period of the binary sequence is in case of Logistic map Equation (6.1) and in case of Equation (6.2) Mapping Every Field Element in GF(2 16 ) to GF(2) using Trace Function The sequence of elements from GF(2 16 ) generated using the chaotic maps as discussed in the previous section, are expressed as powers of g. The elements of GF(2 16 )

128 71 is mapped to GF(2) using Trace function to one bit binary (either 0 or 1) and concatenated. The trace function Tr(β) of any element β Є GF(2 16 ) is given by, (6.4) The arithmetic is carried out modulo (x 16 +x 5 +x 3 +x 2 +1) over GF(2). By definition for O Є GF(2 16 ), Tr(O) = 0 Є GF(2). Hence in case of binary mapping using Trace function, the maximum possible period of the binary sequence is same as that of sequence over GF(2 16 ) in case of Logistic map Equation (6.1) and in case of Equation (6.2). The binary sequence is divided into non-overlapping segments of same length s with different s equal to 15, 31, 63, 127 and 255 and their CCR and LC properties are studied in Chapter 8. The properties of segments of binary sequences derived from chaotic sequence over Z 4 are investigated in the next chapter. 6.5 SUMMARY In this chapter, a scheme for generation of binary sequences using Equations (6.1) and (6.2) defined over GF(2 m ) is explained. In this work two cases, m = 8 and m= 16 are considered to generate chaotic sequences over GF(2 8 ) and GF(2 16 ). These sequences over GF(2 8 ) and GF(2 16 ) are transformed to binary using three techniques.

129 CHAPTER 7 PROPERTIES OF CHAOTIC DISCRETE SEQUENCES OVER Z 4 AND CORRESPONDING BINARY SEQUENCES DERIVED FROM THEM

130 72 CHAPTER 7 PROPERTIES OF CHAOTIC DISCRETE SEQUENCES OVER Z 4 AND CORRESPONDING BINARY SEQUENCES DERIVED FROM THEM This chapter presents the study of Hamming CCR (HCCR) properties and balance property of segments of discrete sequences over Z 4 generated using the model proposed in Chapter 5 and Chapter 6. The CCR properties and LC properties of the segments of binary sequences derived from sequences over Z 4 of same length s, for s = 15, 31, 63, 127 and 255 are investigated. In the next section, the properties of the discrete chaotic sequences over Z 4 are considered. In CDMA application, it is desirable that the set of code sequences has small pairwise peak CCR value or mean square CCR value, which reduces the average BER. Higher linear complexity of code sequences provides inherent security against recovery of data from unauthorized users. 7.1 HAMMING CCR (HCCR) AND BALANCE PROPERTIES OF SEGMENTS OF CHAOTIC DISCRETE SEQUENCES OVER Z 4 As mentioned in Section 5.1, chaotic discrete sequences over Z 4 with six values of multiplication factor n = 5, 6, 7, 8, 9 and 10 are generated. The discrete sequences of different lengths are generated for six chaotic functions given by Equations (7.1) to (7.6). As an example, first discrete values d 0 to d as shown in Figure 7.1 of the generated sequence are considered which is divided into 128 digit non-overlapping segments. The number of non-overlapping segments considered is 100. Each of these segments is numbered as Segment 1, Segment 2 as shown in Figure 7.1. The

131 73 Hamming cyclic CCR as given by Equation (4.10) in Section and balance properties of the generated discrete sequences are investigated. DiscreteSe quence d 0, d 1,... d 127, d 128, d 129,... d 255,... d d Segment1 Segment 2 Segment100 Figure 7.1 Non-overlapping segments of length 128 of discrete sequence over Z HCCR and Balance Properties of Discrete Sequences over Z 4 Generated Using Logistic Map Equations Consider the Logistic map given by Equation (5.1) which is repeated here. where 0 < x < 1 (7.1) A discrete sequence over Z 4 of length is generated using proposed method with bifurcation parameter r = 3.99, initial value x 0 = 0.4 and for each multiplication factor n = 5, 6, 7, 8, 9 and 10. The minimum and maximum values of pairwise normalized cyclic Hamming CCR values of 100 non-overlapping segments of length 128 elements over Z 4 for different values of n are computed using Equation (4.9) and the results are tabulated in Table 7.1. Table 7.1 Hamming Cross Correlation values of Segments of Discrete Sequence of Length 128 Generated using Logistic Map given by Equation (7.1) n value Maximum pairwise Hamming CCR value Minimum pairwise Hamming CCR value

132 Number of occurances of x in a length of For one case, n = 5 and x 0 = 0.4, discrete chaotic sequence of length over Z 4 is generated and histogram is obtained by counting the number of occurrences of elements 0, 1, 2 and 3 in the sequence. It is found that number of occurrences of elements 0, 1, 2 and 3 are 2980, 3240, 3120 and 3460 respectively. The result is plotted as histogram and is shown in Figure 7.2. It is seen that for the generated discrete chaotic sequence of length elements over Z 4 the maximum and minimum number of occurrences differ only by Element value x in Z4 Figure 7.2 Histogram of Discrete Sequence Generated using Logistic Map given by Equation (7.1) Logistic map given by Equation (5.2) is considered next which is repeated here. where -1 < x < 1 (7.2) The real valued sequence generated is with values lying in the range 1 to 1 and after multiplication by n, while taking modulo 4 only positive values are used. To generate discrete sequence over Z 4, initial value x 0 = 0.4, r = 1.99 and six values of n, n = 5, 6, 7, 8, 9 and 10.are considered. For 100 non-overlapping segments of length 128 elements from Z 4, the maximum and minimum pairwise normalized cyclic Hamming

133 Number of occurances of x in a length of CCR values for different values of n are computed using Equation (4.10) and tabulated in Table 7.2. Table 7.2 Hamming Cross Correlation values of Segments of Discrete Sequence of Length 128 Generated using Logistic Map given by Equation (7.2) n value Maximum pairwise Hamming CCR value Minimum pairwise Hamming CCR value Element value x in Z4 Figure 7.3 Histogram of Discrete Sequence Generated using Logistic Map given by Equation (7.2) For obtaining histogram only one case n = 5 and x 0 = 4 is used to generate a chaotic sequence of length elements over Z 4. The number of occurrences of 0, 1, 2 and 3 are counted. The number of occurrences of elements 0, 1, 2 and 3 are 3800, 3050, 2950 and 3000 respectively. Difference of maximum number of occurrences to minimum

134 76 number of occurrences is equal to 850. The result is plotted as histogram as shown in Figure HCCR and Balance Properties of Discrete Sequences over Z 4 Generated Using Tent Map Equation The Tent map equation is considered to generate discrete sequence over Z 4 with initial value x 0 = 0.4 and for each multiplication factor n = 5, 6, 7, 8, 9 and 10. Tent map is given by Equation (5.3) which is repeated here. where 0 < x < 1 (7.3) The normalized cyclic Hamming CCR of 100 non-over-lapping segments of length 128 is computed using Equation (4.10). The initial value x 0 = 0.4 and for each multiplication factor n = 5, 6, 7, 8, 9 and 10 are considered. Table 7.3 gives the maximum and minimum pairwise Hamming cross correlation values. Table 7.3 Hamming Cross Correlation Values of Segments of Discrete Sequence of Length 128 Generated using Tent Map given by Equation (7.3) n value Maximum pairwise Hamming CCR value Minimum pairwise Hamming CCR value To study the balance properties, discrete chaotic sequence of length over Z 4 is generated for one case n = 5 and x 0 = 0.4 and histogram is obtained by counting the number of occurrences of elements 0, 1, 2 and 3 in the sequence. It is seen that number of

135 Number of occurances of x in a length of occurrences of elements 0, 1, 2, and 3 are 3320, 3180, 3120 and 3180 respectively. The histogram is plotted as shown in Figure 7.4. The difference between the maximum number of occurrences to the minimum number of occurrences is 200. This is less than the previous two cases Element value x in Z4 Figure 7.4 Histogram of Discrete Sequence Generated using Tent Map given by Equation (7.3) HCCR and Balance Properties of Discrete Sequences over Z 4 Generated Using Cubic Map Equation The Cubic map is defined by Equation (5.4) which is repeated here. where -1 < x < 1 (7.4) The sequence generated is real sequence with values lying in the range 1 to 1. After multiplication by n, while taking modulo 4 only positive values are used. The initial value x 0 = 0.4 and n = 5, 6, 7, 8, 9 and 10 are considered to generate discrete sequence of length elements over Z 4 for each value of n. The discrete sequence over Z 4 is

136 Number of occurances of x in a length of divided in to 100 non-overlapping segments of length 128 as shown in Figure 7.1. Table 7.4 specifies the maximum and minimum pairwise normalized cyclic Hamming CCR values. Table 7.4 Hamming Cross Correlation Values of Segments of Discrete Sequence of Length 128 Generated using Cubic Map given by Equation (7.4) n value Maximum pairwise Hamming CCR value Minimum pairwise Hamming CCR value For one case, n = 5 and initial value x 0 = 0.4 are considered to generate discrete chaotic sequence of length elements over Z 4 and histogram is obtained by counting the number of occurrences of elements 0, 1, 2 and 3 in the sequence. The number of occurrences of elements 0, 1, 2 and 3 are 3320, 3300, 2920 and 3260 respectively. Difference of maximum number of occurrences to minimum number of occurrences is equal to 400. The result is plotted as histogram and is shown in Figure Element value x in Z4 Figure 7.5 Histogram of Discrete Sequence Generated using Cubic Map given by Equation (7.4)

137 HCCR and Balance Properties of Discrete Sequences over Z 4 Generated Using Quadratic Map Equation Consider Quadratic map defined by Equation (5.5) which is repeated here. where < x < 0.5 (7.5) Sequence element x lies in the range 0.5 to 0.5. After multiplying each elements of sequence {x k } by n, while taking modulo 4 only positive values are considered. Initial value x 0 = 0.4, r = 0.4 and n = 5, 6, 7, 8, 9 and 10 are considered to generate chaotic discrete sequences of length elements over Z 4 for each value of n. The generated discrete sequence is divided in to 100 non-overlapping segments of length 128 over Z 4. The maximum and minimum pairwise normalized cyclic Hamming cross correlation values is tabulated in Table 7.5. Table 7.5 Hamming Cross Correlation Values of Segments of Discrete Sequence of Length 128 Generated using Quadratic Map given by Equation (7.5) n value Maximum pairwise Hamming CCR value Minimum pairwise Hamming CCR value

138 Number of occurances of x in a length of Element value x in Z4 Figure 7.6 Histogram of Discrete Sequence Generated using Quadratic Map given by Equation (7.5) To study the balance properties, the number of occurrences of 0, 1, 2 and 3 in the sequence of length are counted for n = 5 and x 0 = 0.4 and plotted as histogram in Figure 7.6. It is seen that number of occurrences of elements 0, 1, 2, and 3 are 3330, 2850, 3200 and 3420 respectively. The difference between the maximum number of occurrences to the minimum number of occurrences is HCCR and Balance Properties of Discrete Sequences over Z 4 Generated Using Bernoulli Map Equation To study the pairwise normalized cyclic Hamming CCR properties and balance property, chaotic discrete sequences of length are generated with the initial value x 0 = 0.4, r = 1.99 and for each multiplication factor n = 5, 6, 7, 8, 9 and 10 using Bernoulli map equation. Bernoulli map is given by Equation (5.1) which is repeated here.

139 81 where < x < 0.5 (7.6) The sequence generated is real sequence with values lying in the range 0.5 to 0.5. After multiplication by n, while taking modulo 4 only positive values are considered for each value of n. The maximum and minimum pairwise normalized cyclic Hamming CCR value of 100 non-overlapping segments of length 128 is tabulated in Table 7.6. To obtain histogram, one discrete chaotic sequence of length over Z 4 is generated with n = 5, r = 1.99 and x 0 = 0.4. The number of occurrences of elements 0, 1, 2 and 3 in the sequence is counted and found to be 3150, 3220, 3250 and 3180 respectively. The result is plotted as histogram and is shown in Figure 7.7. Difference of maximum number of occurrences to minimum number of occurrences is equal to 100. The difference is least compared to other cases. Thus excellent balance property is exhibited by discrete sequence over Z 4 using Bernoulli map with n = 5 and x 0 = 0.4. Table 7.6 Hamming Cross Correlation Values of Segments of Discrete Sequence of Length 128 Generated using Bernoulli Map given by Equation (7.6) n value Maximum pairwise Hamming CCR value Minimum pairwise Hamming CCR value

140 Number of occurances of x in a length of Element value x in Z4 Figure 7.7 Histogram of Discrete Sequence Generated using Bernoulli Map given by Equation (7.6) From Tables 7.1, 7.2, 7.3, 7.4, 7.5 and 7.6 it is seen that HCCR property exhibited by discrete sequence over Z 4 obtained using different chaotic maps except Quadratic map, are almost identical for values of n 5 < n < 10. Excellent balance property is exhibited by discrete sequence over Z 4 generated using Tent map and Bernoulli map for the chosen values of n = 5 and x 0 = 0.4 having difference of maximum number of occurrences to minimum number of occurrences equal to 200 and 100 respectively. Having discussed the normalized cyclic Hamming cross correlation (HCCR) and balance properties of chaotic discrete sequences over Z 4, in the next section, the properties of binary sequences derived from discrete chaotic sequence over Z 4 are considered. As discussed in Section 1.4, four methods of deriving binary sequences from

141 83 discrete sequence over Z 4 are considered. The first method using mapping P 0 is discussed next. 7.2 CCR PROPERTIES OF SEGMENTS OF CHAOTIC BINARY SEQUENCES DERIVED FROM SEQUENCE OVER Z 4 USING MAPPING P 0 For each of the six chaotic map equations considered, real valued sequence {x k }, discrete sequence {y k } over Z 4 is generated by choosing the initial value x 0 = 0.4 and for six values of multiplication factor n = 5, 6, 7, 8, 9 and 10. This discrete sequence {y k } is converted to binary using four methods as discussed in Section 5.2 and 5.3 To study the CCR properties of the binary sequence, non-overlapping segments of same length s, for s = 15, 31, 63, 127 and 255 are considered. For each value of multiplication factor n = 5, 6, 7, 8, 9, 10 considered, 100 segments of length 15, 100 segments of length 31, 200 segments of length 63, 300 segments of length 127 and 700 segments of length 255 as listed in Table 1.1 are considered. CCR and LC properties of these sequences are investigated. Out of Λ segments of binary sequences of same length as given in Table 1.1, segments having magnitude of pairwise CCR value less than or equal to α max are selected. Other segments are discarded. For the sake of comparison of segments of binary sequences generated with conventional code sequences like Gold and large set Kasami sequences of same length, the maximum magnitude of pairwise CCR value denoted by α max is taken to be same as that of Gold and large set Kasami sequences of same length. The maximum magnitude of pairwise CCR value of Gold and large set Kasami sequences of different length are given in Table 7.7. To check whether sequences with lower pairwise CCR value than Gold sequences are present, two more values of α max which is less than that of Gold and large set Kasami sequences of same length are considered. It is seen that if α max is decreased, number of segments of sequences having pairwise CCR value less than or equal to α max also decreases. It is also seen that the number of segments of sequences having peak magnitude of pairwise CCR value same as

142 84 that of Gold and large set Kasami sequences of same length, is more than the number of Gold sequences. The number of possible Gold and Kasami sequences of small set and large set of different lengths with their magnitude of pairwise CCR value and LC are tabulated in Table 7.7 (John G. Proakis 2001). Kasami sequences do not exist for odd number of stages and hence there are no Kasami sequences of length 31, 127 etc. Small set Kasami sequences have better cross correlation properties compared to Gold sequences (Sarwate D. V. et al 1980). However, the small set Kasami sequences contain number of sequences γ = 2 L/2. Large set Kasami sequences contains number of sequences γ = 2 L/2 (2 L +1) and the cross correlation function is five valued. The magnitude of pairwise CCR value in case of large set Kasami sequences is same as that of Gold sequences. Also the LC of large set Kasami sequences is more than that of Gold sequences and the LC of small set Kasami sequences is less than that of Gold sequences (Andrew Klapper 1995). Eventhough number of large set Kasami sequences is more than number of Gold sequences having same peak magnitude of CCR values, there are no large set Kasami sequences of length (2 L - 1) for L odd. In the proposed scheme of generation of binary sequences it is possible to have number of segments of sequences of length 31 and 127 more than the number of Gold sequences of same length. No. of stages L Table 7.7 Length of Sequence, Maximum Value of Pairwise CCR Value α max, Number of Sequences γ and LC of Gold and Kasami Sequences Sequence length s =2 L -1 Gold Sequences α max γ = 2 L +1 LC = 2L α max Small Set γ = 2 L/2 Kasami Sequences Large Set LC α max γ = = 3L/2 2 L/2 (2 L +1) LC = 5L/

143 85 Segments of sequences, having maximum magnitude of pairwise CCR value same as that of Gold and large set Kasami sequences of same length are selected. Also segments of sequences having two lower values of pairwise CCR value which is less than that of Gold and large set Kasami sequences of same length are selected. The length of segment, maximum magnitude of pairwise CCR value of Gold or large set Kasami sequences and two lower values of CCR values selected are listed in Table 7.8. It is to be noted that there are no Gold or Kasami sequences of given length having magnitude of pairwise CCR value less than listed in column 3 and 4 in Table 7.8. Length of Segment s Table 7.8 Length of Segments and the Corresponding Maximum Pairwise CCR values Chosen Maximum magnitude of pairwise CCR value of Gold or large set Kasami Sequences Two other values of pairwise CCR values less than that of Gold or large set Kasami Sequences Number of non-overlapping segments of different lengths generated using six chaotic map Equations (7.1) to (7.6) having magnitude of pairwise CCR value less than or equal to α, for three different values of α, one corresponding to that of Gold sequences and two lower values are tabulated in Tables 7.9 to Table 7.9 lists the selected number of segments of binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.1) and using mapping P 0. For segments of same length s, for s = 15, 31, 63, 127 and 255, three values of α is considered for each length. First column gives the trial number. Second column gives the value of n chosen. The number of sequences of length 15 having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 respectively are given in the next three columns. Magnitude of pairwise CCR values for set of segments of same length s, for s = 31, 63, 127 and 255 are determined. The number of segments of length s having magnitude of pairwise CCR value less than three particular values are listed in the remaining columns.

144 86 Details of trial number 1 for segments of sequences of length 15 are given in first row of Table 7.9. The number of non-overlapping segments of sequences of length 15 considered in this work is 100 as listed in Table 1.1. For n = 5 out of 100 segments, 7 segments have magnitude of pairwise CCR value 0.35, 15 segments have magnitude of pairwise CCR value 0.5 and 28 segments have magnitude of pairwise CCR value 0.6. Gold sequences of length 15 have pairwise CCR value 0.6 and there are only 17 sequences. Likewise the number of segments selected for other set of sequences of length s, for s = 31, 63, 127 and 255 with three values of α are listed in remaining columns. The number of segments selected for Trial numbers 2 to 6 for n = 6, 7, 8, 9 and 10 respectively are also tabulated. It is seen from Table 7.9 that, number of segments of sequences of length 15 having magnitude of pairwise CCR value less than or equal to α decreases for lower value of α. This is true for other segments of same length s, for s = 31, 63, 127 and 255. Table 7.9 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Trial No n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.10 gives the number of non-overlapping segments of sequences derived from sequence over Z 4 defined by Logistic map Equation (7.2) and using mapping P 0. The segments of same length s, for s = 15, 31, 63, 127 and 255 are considered. The number of segments of sequences having magnitude of pairwise CCR values same as that of Gold sequences is also given. It is seen that this number is more than the number of Gold sequences of same length given in Table 7.7.

145 87 Table 7.10 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.11, 7.12, 7.13 and 7.14 gives the number of segments of same length s, for s = 15, 31, 63, 127 and 255 derived from chaotic sequences over Z 4 defined by Tent map given by Equation (7.3), Cubic map given by Equation (7.4), Quadratic map given by Equation (7.5) and Bernoulli map given by Equation (7.6) respectively and using mapping P 0, having magnitude of pairwise CCR value α. Table 7.11 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

146 88 Table 7.12 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.13 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.14 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Mapping P 0 having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

147 89 There are 17 Gold sequences of length 15 having magnitude of pairwise CCR value 0.6. The binary sequences derived from sequence over Z 4, have more than 17 segments of length 15 with magnitude of pairwise CCR value less than 0.6. For Example in Table 7.9 for n = 5 there are 28 segments of sequences, in Table 7.10 for n = 6 and 10 there are 28 sequences. From Tables 7.9 to 7.14, it is seen that the number of segments of chaotic binary sequences having magnitude of pairwise CCR value same as that of Gold sequences, is more than the number of Gold sequences of same length. Also there are some segments of sequences with magnitude of pairwise CCR value less than that of Gold sequences. Table 7.15 Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Mapping P 0 Length of Sequence Value of n Chaotic Map Equation and Table Number Maximum Number of Sequences Quadratic Map, Table Cubic Map, Table Logistic Map Equation (7.1), Table 7.9 6, 9 Logistic Map Equation (7.2), Table , 8 Tent Map Equation (7.3), Table , 8 Cubic Map Equation (7.4), Table , 9 Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Tent Map Equation (7.3), Table Cubic Map Equation (7.4), Table Logistic Map Equation (7.2), Table Cubic Map Equation (7.4), Table Quadratic Map Equation (7.5), Table , 10 Bernoulli Map Equation (7.6), Table Bernoulli Map Equation (7.6), Table Cubic Map Equation (7.4), Table Bernoulli Map Equation (7.6), Table Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.1), Table Bernoulli Map Equation (7.6), Table Bernoulli Map Equation (7.6), Table Bernoulli Map Equation (7.6), Table Cubic Map Equation (7.4), Table Quadratic Map Equation (7.5), Table

148 90 Table 7.15 lists the summary of Tables 7.9 to It gives the choice of chaotic map with x 0 = 0.4 which gives maximum number of segments of binary sequences derived from sequence over Z 4 and of different length s, having peak magnitude of pairwise CCR value less than or equal to three chosen values of α. First column in Table 7.15 gives the length of the sequence, second column gives three values of magnitude of pairwise CCR value chosen, third column gives values of n, fourth column gives chaotic map and corresponding Table number from which maximum number of segments of sequences having magnitude of pairwise CCR value less than α are considered and is given in the last column. From the trials carried out and values listed in Tables (7. 9) to (7. 14), it is seen that for length 15, i) Maximum number of segments having same CCR as Gold sequence is equal to 35 for n = 9 and Quadratic map as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 25 for n = 6 and Cubic map as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 9 atleast for one value of n in all cases of chaotic maps. For length 31, i) Maximum number of segments having same CCR as Gold sequence is equal to 50 for n = 5 and Tent map as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.23 is equal to 25 for n = 8 and Cubic map as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 15 atleast for one value of n in all cases of chaotic maps. For length 63, i) Maximum number of segments having same CCR as Gold sequence is equal to and is 137 for n = 6 and Bernoulli map as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 49 for n = 10 and Cubic map as given in Table 7.12.

149 91 iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 30 for n = 10 and Quadratic Map Equation (7.5) as given in Table 7.13 and Bernoulli map Equation (7.6) as given in Table For length 127, i) Maximum number of segments having same CCR as Gold sequence is equal to and is 222 for n = 5 and Logistic Map Equation (7.1) as given in Table 7.9. ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 81 for n = 8 and Bernoulli Map Equation (7.6) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 41 for n = 5 and Bernoulli Map Equation (7.6) as given in Table For length 255, i) Maximum number of segments having same CCR as Gold sequence is equal to and is 431 for n = 5 and Bernoulli Map Equation (7.6) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 135 for n = 5 and Cubic map as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 71 for n = 6 and Quadratic Map Equation (7.5) as given in Table From Table 7.15, by properly choosing n and chaotic map, it is possible to obtain large number of segments of sequences of given length whose magnitude of pairwise CCR value is same as that of Gold or large set Kasami sequences. The number of such segments is almost double that of number of Gold sequences of same length. It is possible to obtain set of sequences having magnitude of pairwise CCR value is less than that of Gold or large set Kasami sequences.

150 92 Having discussed the CCR properties of segments of binary sequences derived from sequence over Z 4 defined by all the six chaotic functions, in the next section, the linear complexity of 15 bit non-overlapping segments having desirable pairwise CCR values for six chaotic map equations are discussed. 7.3 LINEAR COMPLEXITY PROPERTIES OF SEGMENTS OF BINARY SEQUENCES OF LENGTH 15 DERIVED FROM SEQUENCE OVER Z 4 USING MAPPING P 0 Based on the magnitude of pairwise CCR value α, segments of sequences are selected for three values of α, 0.6 corresponding to peak magnitude of CCR value of Gold sequences and large set Kasami sequences and two smaller values for which there are no Gold sequences or large set Kasami sequences. Segments of sequences of length s = 15 having magnitude of pairwise CCR value same as that of Gold sequences and large set Kasami sequences are selected. The binary sequences are obtained using proposed method of deriving binary sequences from sequence over Z 4 defined by all the six chaotic map equations and using mapping P 0. These segments of sequences of length 15 are investigated for LC properties. The 15 segments having 0.6 which are selected out of 100 segments are as listed in Tables 7.9 to 7.14 for all the six chaotic functions defined by Equations (7.1) to (7.6), LC of these sequences are computed. LC is also computed for segments of sequences of same length s, for s = 31, 63, 127 and 255 and it is observed by computer search that LC is greater than that of Gold sequences and large set Kasami sequences of same length. In what follows it is shown that, it is possible to get some segments of sequences of length s = 15 with LC more than that of Gold sequences of same length. In the next section, LC of segments of sequences derived from sequence over Z 4 using Logistic map defined by Equations (7.1) and (7.2) are considered.

151 LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Mapping P 0 Binary sequences derived from discrete sequence over Z 4 defined by Logistic map given by Equation (7.1) with initial value x 0 = 0.4, r = 3.99, with n = 5 and using mapping P 0 is considered first. Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 7.9 is considered and their LC is computed. For n = 5 out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 15 segments have magnitude of pairwise CCR value 0.5. The LC of these 15 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 28 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 28 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. It is to be noted that there are no Gold sequences of length 15 having magnitude of pairwise CCR value 0.35 or 0.5. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 11. Only sequences corresponding to n= 5 are listed.

152 94 Table 7.16 Linear complexity of 7 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.17 Linear complexity of 15 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.18 Linear complexity of 28 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

153 95 Table 7.18 (Continued) The LC of Gold sequence of length 15 as listed in Table 7.7, is 8 and that of large set of Kasami sequences is 10. Chaotic binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.1) and using mapping P 0, it is possible to obtain some segments of sequences of length s = 15 having LC greater than that of Gold sequences and magnitude of pairwise CCR value less than that of Gold sequences. LC properties of segments of sequences generated using Logistic map given by Equation (7.2) is discussed next. Consider segments of binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.2) with initial value x 0 = 0.4, bifurcation parameter r = 1.99 and using mapping P0. Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 7.10 is considered and their LC is computed. For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table 7.19.

154 96 b) 16 segments have magnitude of pairwise CCR value 0.5. The LC of these 16 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 24 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 24 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. Table 7.19 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.20 Linear complexity of 16 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

155 97 Table 7.21 Linear complexity of 24 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity With n = 6, 7, 8, 9 and 10, the corresponding sequences of length 15 also found to have LC between 8 and 11 which are not listed. The LC of some segments of sequences of length s = 15 is greater than that of Gold sequences of same length LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map Equation and Mapping P0 The segments of binary sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.11 are considered and their LC is computed.

156 98 For n = 5 out of 100 segments of length 15, a) 9 segments have magnitude of pairwise CCR value The LC of these 9 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 17 segments have magnitude of pairwise CCR value 0.5. The LC of these 17 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 24 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 24 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. Table 7.22 Linear complexity of 9 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

157 99 Table 7.23 Linear complexity of 17 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table7.24 Linear complexity of 24 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

158 100 For segments of length 15 in case of n = 6, 7, 8, 9 and 10 which are not listed here, the LC varies between 8 and 10. In this case also the LC of some segments of length s = 15 is more than that of Gold sequences of same length LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map Equation and Mapping P 0 The linear complexity of segments of sequences of length s = 15 derived from sequence over Z 4 defined by cubic map given by Equation (7.4) and using mapping P 0 having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.12 are computed. For n = 5 out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 15 segments have magnitude of pairwise CCR value 0.5. The LC of these 15 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 31 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 31 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10.

159 101 Table 7.25 Linear complexity of 7 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.26 Linear complexity of 15 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.27 Linear complexity of 31 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

160 102 Table 7.27 (Continued) LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Mapping P 0 Binary sequences derived from discrete sequence over Z 4 defined by Quadratic map given by Equation (7.5) with initial value x 0 = 0.4, r = 0.4, with n = 5 and using mapping P 0 is considered. Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 7.13 is considered and their LC is computed.

161 103 For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 13 segments have magnitude of pairwise CCR value 0.5. The LC of these 13 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 30 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 30 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. Table 7.28 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.29 Linear complexity of 13 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

162 104 Table 7.29 (Continued) Table 7.30 Linear complexity of 30 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

163 LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Mapping P 0 Binary sequences derived from discrete sequence over Z 4 defined by Bernoulli map given by Equation (7.6) with initial value x 0 = 0.4, r = 1.99, with n = 5 and using mapping P 0 is considered. Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 7.14 is considered and their LC is computed. For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 16 segments have magnitude of pairwise CCR value 0.5. The LC of these 16 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 29 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 29 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. Table 7.31 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

164 106 Table 7.32 Linear complexity of 16 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.33 Linear complexity of 32 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Mapping P 0 having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

165 107 Table 7.33 (Continued) As in the case of binary sequences of length 15 derived from sequences over Z 4 based on other map equations, in this case also linear complexity of some segments is more than that of Gold sequences of same length. Having discussed the CCR and LC properties of binary sequences derived from sequences over Z 4 using mapping P 0 for all the six chaotic functions, the properties of binary sequences obtained using polynomial mapping discussed in Section 5.3 are considered next. The three polynomial mappings (Udaya P. and Siddiqi M.U. 1996) given by Equations (1.9) to (1.11) and are repeated here, i) (7.7) ii) (7.8) iii) (7.9) Deriving chaotic binary sequences from sequence over Z 4 defined by all six chaotic maps and using polynomial mapping defined by Equation (7.7) and their CCR properties are considered in the next section.

166 CCR PROPERTIES OF CHAOTIC BINARY SEQUENCES DERIVED FROM SEQUENCE OVER Z 4 USING POLYNOMIAL MAPPING For each of the six chaotic functions defined by Equations (7.1) to (7.6), real valued sequence {x k } and discrete sequence {y k } over Z 4 is derived by choosing the initial value x 0 = 0.4 and multiplication factor n = 5, 6, 7, 8, 9 and 10. This discrete sequence {y k } is converted to binary as discussed in Section 5.3 using polynomial mapping given by Equation (7.7). To study the properties of the generated binary sequence, non-overlapping segments of same length s, for s = 15, 31, 63, 127 and 255 are considered. The number of non-overlapping segments of different lengths considered in this work is listed in Table 1.1. The number of segments having magnitude of pairwise CCR values less than or equal to α is determined. Gold and large set Kasami sequences of length 15 have magnitude of pairwise CCR value 0.6, sequences of length 31 have magnitude of pairwise CCR value , sequences of length 63 have magnitude of pairwise CCR value , sequences of length 127 have magnitude of pairwise CCR value and sequences of length 255 have magnitude of pairwise CCR value as given in Table 7.7. The number of non-overlapping segments of different lengths having magnitude of pairwise CCR value less than or equal to α max corresponding to Gold sequence and two values less than this value are determined from computer search. Segment of binary sequences having magnitude of pairwise CCR value less than or equal to α, for three values of α are considered for all the cases of six chaotic functions defined by Equations (7.1) to (7.6) and are tabulated in Tables 7.34 to Details of trial number 1 for segments of sequences of length 15 are given in first row of Table The number of non-overlapping segments of sequences of length 15 considered in this work is 100 as listed in Table 1.1. For n = 5 out of 100 segments, 9 segments have magnitude of pairwise CCR value 0.35, 19 segments have magnitude of pairwise CCR value 0.5 and 29 segments have magnitude of pairwise CCR value 0.6. Likewise the number of segments selected for other set of sequences of length s,

167 109 for s = 31, 63, 127 and 255 with three values of α are listed in remaining columns. The number of segments selected for Trial numbers 2 to 6 for n = 6, 7, 8, 9 and 10 respectively are also tabulated. It is seen from Table 7.34 that, number of segments of sequences of length 15 having magnitude of pairwise CCR value less than or equal to α decreases for lower value of α. This is true for other segments of same length s, for s = 31, 63, 127 and 255. Table 7.34 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Trial No. Table 7.35 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

168 110 Table 7.36 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.37 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.38 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

169 111 Trial No. Table 7.39 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Polynomial Mapping P 1 (y) having Different Magnitude of Pairwise CCR Value n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.40 lists the summary of Tables 7.34 to 7.39 which gives the maximum number of segments of sequences of same length s, for s = 15, 31, 63, 127 and 255 and whose magnitude of pairwise CCR value is less than three particular values chosen. Table 7.40 Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping P 1 (y) Length of Sequence Value of n Chaotic Map Equation and Table Number Maximum Number of Sequences 0.6 6, 8 Logistic Map Equation (7.2), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.2), Table Tent Map Equation (7.3), Table , 7 Cubic Map Equation (7.4), Table Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.1), Table Logistic Map Equation (7.2), Table Logistic Map Equation (7.2), Table Tent Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.1), Table Cubic Map Equation (7.4), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.1), Table Logistic Map Equation (7.2), Table Cubic Map Equation (7.5), Table Tent Map Equation (7.3), Table Cubic Map Equation (7.4), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.2), Table Logistic Map Equation (7.2), Table , 6 Cubic Map Equation (7.4), Table

170 112 From the trials carried out and values listed in Tables (7.34) to (7.39), it is seen that for length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 41 for n = 6, 8 and Logistic Map Equation (7.2) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 26 for n = 8 and Bernoulli Map Equation (7.6) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 10 atleast for one value of n in all cases of chaotic maps except Logistic map Equation (7.1). For length 31, i) Maximum number of segments having same CCR as Gold is equal to and is 56 for n = 8 and Logistic Map Equation (7.1) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.23 is equal to 29 for n = 5 in Logistic Map Equation (7.2) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 16 with n = 5 using Logistic Map Equation (7.2) as given in Table 7.35, or with n = 6 using Tent map as given in Table 7.34 and also with n = 10 using Bernoulli map as given in Equation For length 63, i) Maximum number of segments having same CCR as Gold is equal to and is 138 for n = 7 and Logistic Map Equation (7.1) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 49 for n = 5 using Cubic map as given in Table 7.37 and also Bernoulli map as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 31 for n = 10 using Logistic Map Equation (7.1) as given in Table 7.34 or with n = 5 using Logistic Map Equation (7.2) as given in Table 7.35 and also for n = 8 using Cubic map as given in Table 7.37.

171 113 For length 127, i) Maximum number of segments having same CCR as Gold is equal to and is 246 for n = 9 and Tent Map Equation (7.3) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 86 for n = 6 and Cubic Map Equation (7.4) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 45 for n = 8 and Bernoulli Map Equation (7.6) as given in Table For length 255, i) Maximum number of segments having same CCR as Gold is equal to and is 439 for n = 6 and Logistic Map Equation (7.2) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 131 for n = 6 in Logistic Map Equation (7.2) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 85 for n = 5 or 6 and Cubic Map Equation (7.4). The number of segments of binary sequences derived from sequence over Z 4 defined by chaotic map Equations (7.1) to (7.6) and using polynomial mapping P 1 (y) given by Equation (7.7) of same length s, for s = 15, 31, 63, 127 and 255 having magnitude of pairwise CCR value same as that of Gold sequences, is found to be more than the number of Gold sequences of same length. The number of Gold and large set Kasami sequencs of different lengths along with their peak value of magnitude of pairwise CCR value is given in Table 7.7. Also there are some segments of sequences with magnitude of pairwise CCR value less than that of Gold sequences and large set Kasami sequences of same length. However number of such sequences is less than the number of Gold sequences and large set Kasami sequences of same length. The LC of segments of binary sequences of length 15 derived from sequences over Z 4 defined by all six chaotic functions given by Equations (7.1) to (7.6) and using polynomial mapping P 1 (y) given by Equation (7.7) is considered in the next section.

172 LINEAR COMPLEXITY PROPERTIES OF SEGMENTS OF BINARY SEQUENCES OF LENGTH 15 DERIVED FROM SEQUENCE OVER Z 4 USING POLYNOMIAL MAPPING The binary sequences are obtained using proposed method of deriving binary sequences from sequence over Z 4 defined by all the six chaotic map equations and using polynomial mapping P 1 (y) given by Equation (7.7). Based on the magnitude of pairwise CCR value α, segments of sequences are selected for three values of α. 0.6 corresponding to peak magnitude of CCR value of Gold sequences and large set Kasami sequences and two smaller values for which there are no Gold sequences or large set Kasami sequences. These segments of sequences of length 15 are investigated for LC properties. The segments of length 15, having 0.6 which are selected are as listed in Tables 7.34 to 7.39 for all the six chaotic functions defined by Equations (7.1) to (7.6), LC of these sequences are computed. LC is also computed for segments of sequences of same length s, for s = 31, 63, 127 and 255 and it is observed by computer search that LC is greater than that of Gold sequences and large set Kasami sequences of same length. Study of LC properties of segments of binary sequences derived from sequence over Z 4 defined by Logistic map equations and using polynomial mapping P 1 (y)is considered in the next section LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Using Polynomial Mapping Segments of binary sequences derived from chaotic discrete sequences over Z 4 defined by Logistic map given by Equation (7.1) with initial value x 0 = 0.4, r = 3.99, n = 5 and using polynomial mapping P 1 (y) is considered first. Segments of sequences of length 15 having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as tabulated in Table 7.34 is considered and their LC is computed.

173 115 For n = 5 out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 15 segments have magnitude of pairwise CCR value 0.5. The LC of these 15 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 29 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 29 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. Table 7.41 Linear complexity of 9 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.42 Linear complexity of 19 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

174 116 Table 7.42 (Continued) Table 7.43 Linear complexity of 29 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

175 117 For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. LC properties of segments of sequences generated using Logistic map given by Equation (7.2) is discussed next. Consider segments of binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.2) with initial value x 0 = 0.4, bifurcation parameter r = 1.99 and using polynomial mapping P 1 (y)given by Equation (7.7). The number of segments of sequences out of 100 segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as listed in Table 7.35 is considered to study the linear complexity properties. For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 22 segments have magnitude of pairwise CCR value 0.5. The LC of these 22 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 32 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 32 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 12. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 11. Only sequences corresponding to n= 5 are listed.

176 118 Table 7.44 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.45 Linear complexity of 22 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

177 119 Table 7.46 Linear complexity of 32 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map Equation and Using Polynomial Mapping Tent map given by Equation (7.3) is considered with initial value x 0 = 0.4 to derive sequences over Z 4. These binary sequences are mapped to binary using polynomial mapping P 1 (x). As in the earlier cases the out of 100 segments, number of segments of sequences having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.36 are considered.

178 120 For n = 5 out of 100 segments of length 15, a) 10 segments have magnitude of pairwise CCR value The LC of these 10 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 23 segments have magnitude of pairwise CCR value 0.5. The LC of these 23 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 31 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 31 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 11. Only sequences corresponding to n= 5 are listed. Table 7.47 Linear complexity of 10 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

179 121 Table 7.48 Linear complexity of 23 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table7.49 Linear complexity of 31 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

180 122 Table 7.49 (Continued) LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map Equation and Using Polynomial Mapping Cubic map given by Equation (7.4) is considered next to generate sequence over Z 4 with initial value x 0 = 0.4 and n = 5. As in the earlier cases binary sequence is derived by using polynomial mapping P 1 (y) and the number of segments which are selected out of 100 segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.37 is considered. For n = 5 out of 100 segments of length 15, a) 10 segments have magnitude of pairwise CCR value The LC of these 10 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 25 segments have magnitude of pairwise CCR value 0.5. The LC of these 25 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 33 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 33 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10.

181 123 For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.50 Linear complexity of 10 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.51 Linear complexity of 25 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

182 124 Table 7.52 Linear complexity of 33 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

183 LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Using Polynomial Mapping Quadratic map given by Equation (7.5) is considered next with initial value x 0 = 0.4 n = 5 and r = 0.4 to generate binary sequence from sequence over Z 4 using polynomial mapping P 1 (y). Out of 100 segments, number of segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 is listed in Table 7.38 is considered to compute the LC. For n = 5 out of 100 segments of length 15, a) 9 segments have magnitude of pairwise CCR value The LC of these 9 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 23 segments have magnitude of pairwise CCR value 0.5. The LC of these 23 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 32 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 32 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. It is to be noted that there are no Gold sequences of length 15 having magnitude of pairwise CCR value 0.35 or 0.5. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed.

184 126 Table 7.53 Linear complexity of 9 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.54 Linear complexity of 23 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

185 127 Table 7.55 Linear complexity of 32 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Using Polynomial Mapping Bernoulli map given by Equation (7.6) is considered to generate sequence over Z 4 with initial value x 0 = 0.4, n = 5, r = Binary sequences are derived by using

186 128 polynomial mapping P 1 (y). Out of 100 segments, number of segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 is listed in Table 7.39 is considered to compute the LC. For n = 5 out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 20 segments have magnitude of pairwise CCR value 0.5. The LC of these 20 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 37 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 37 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.56 Linear complexity of 7 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

187 129 Table 7.57 Linear complexity of 20 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.58 Linear complexity of 37 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 1 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

188 130 Table 7.58 (Continued) Having discussed the CCR and LC properties of binary sequences derived from sequences over Z 4 defined by all six chaotic functions and using polynomial mapping P 1 (y), CCR properties of segments of binary sequences obtained using polynomial mapping P 2 (y) is considered next. 7.6 CCR PROPERTIES OF CHAOTIC BINARY SEQUENCES DERIVED FROM SEQUENCE OVER Z 4 USING POLYNOMIAL MAPPING For each of the six chaotic functinons considered defined by Equations (7.1) to (7.6), real valued sequence {x k } and discrete sequence {y k } over Z 4 is generated by choosing the initial value x 0 = 0.4 and six values of multiplication factor n = 5, 6, 7, 8, 9 and 10 as discussed in Section 5.1. This discrete sequence {y k } is converted to binary as discussed in Section 5.3 using polynomial mapping given by Equation (7.8). To study the pairwise CCR and LC properties, non-overlapping segments

189 131 of same length s, for s = 15, 31, 63, 127 and 255 are considered. The number of segments considered for same length s, for s = 15, 31, 63, 127 and 255 are listed in Table 1.1. The number of segments of same length s, for s = 15, 31, 63, 127 and 255 having different magnitude of pairwise CCR values is determined and tabulated in Tables 7.59 to Table 7.59 gives the number of non-overlapping segments of different lengths derived from sequence over Z 4 defined by Logistic map given by Equation (7.1) and using polynomial mapping P 2 (y). It is seen that the number of segments of length 15 is more than number of Gold sequences of same length, having magnitude of pairwsise CCR value 0.6, corresponding to that of Gold sequences of same length.. In this case also there are some segments of sequences less than the number of Gold and large set Kasami sequences of same length, with magnitude of pairwise CCR value α less than that of Gold and large set Kasami sequences. Almost similar results are seen for other chaotic functions defined by Equations (7.2) to (7.6) as tabulated in Tables 7.60 to 7.64 respectively. Table 7.59 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

190 132 Table 7.60 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.61 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.62 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

191 133 Table 7.63 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.64 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Polynomial Mapping P 2 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Summary of Tables 7.59, 7.60, 7.61, 7.62, 7.63, and 7.64 is given in Table Table 7.65 gives the details of length of sequence in first column, three values of magnitude of pairwise CCR value chosen in second column, value of n in third column, chaotic map and table number in fourth column and maximum number of segments of sequences in the last column. From the trials carried out and values listed in Tables (7.59) to (7.64), it is seen that for length 15,

192 134 i) Maximum number of segments having same CCR as Gold is equal to and is 39 for n = 8 and Logistic Map Equation (7.1) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 25 for n = 5 using Logistic Map Equation (7.2) as given in Table 7.60 and for n = 8 using Tent Map Equation (7.3) as given in Table 7.61 and also for n = 8 using Quadratic Map Equation (7.5) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 13 for n = 6 using Tent Map Equation (7.3) as given in Table 7.61 or Quadratic Map Equation (7.5) as given in Table Table 7.65 Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping P 2 (y) Length of Sequence Value of n Chaotic Map Equation and Table Number Maximum Number of Sequences Logistic Map Equation (7.1), Table Logistic Map Equation (7.2), Table Tent Map Equation (7.3), Table Quadratic Map Equation (7.5), Table Tent Map Equation (7.3), Table Quadratic Map Equation (7.5), Table Logistic Map Equation (7.2), Table Logistic Map Equation (7.2), Table or 10 Logistic Map Equation (7.2), Table Cubic Map Equation (7.4), Table Logistic Map Equation (7.2), Table Tent Map Equation (7.3), Table Quadratic Map Equation (7.5), Table Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.2), Table Quadratic Map Equation (7.5), Table Logistic Map Equation (7.1), Table Cubic Map Equation (7.4), Table Bernoulli Map Equation (7.6), Table Tent Map Equation (7.3), Table Logistic Map Equation (7.1), Table From Table 7.65 it also seen that for length 31, i) Maximum number of segments having same CCR as Gold is equal to and is 60 for n = 6 and Logistic Map Equation (7.2) as given in Table 7.60.

193 135 ii) Maximum number of segments having magnitude of pairwise CCR value 0.23 is equal to 29 for n = 6 using Logistic Map Equation (7.2) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 16 for n = 6 or 10 using Logistic Map Equation (7.2) as given in Table For length 63, i) Maximum number of segments having same CCR as Gold is equal to and is 136 for n = 7 and Cubic Map Equation (7.4) as given by Table 7.62 ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 49 with n = 7 using Logistic Map Equation (7.2) as given by Table 7.60 or Quadratic Map Equation (7.5) as given by Table 7.63 and also with n = 6 using Tent Map Equation (7.3) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 27 with n = 8 using Quadratic Map Equation (7.5) as given by Table For length 127, i) Maximum number of segments having same CCR as Gold is equal to and is 256 for n = 6 and Bernoulli Map Equation (7.6) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 89 with n = 7 using Logistic Map Equation (7.2) as given by Table 7.60 or Quadratic Map Equation (7.5) as given by Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 49 with n = 5 using Logistic Map Equation (7.1) as given in Table 7.59 and also with n = 10 using Cubic map as given by Table For length 255, i) Maximum number of segments having same CCR as Gold is equal to and is 488 for n = 5 and Bernoulli Map Equation (7.6) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 150 for n = 9 and Tent Map Equation (7.3) as given in Table 7.61.

194 136 iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 89 for n = 10 and Logistic Map Equation (7.1) as given in Table In this case also, the number of segments of sequences derived from sequence over Z 4 defined by chaotic maps given by Equations (7.1) to (7.6) with initial value x 0 = 0.4 and n = 5, 6, 7, 8, 9 and 10 and using polynomial mapping P 2 (y), is more than the number of Gold sequences of same length with magnitude of pairwise CCR value 0.6. Also there are some segements of sequences having magnitude of pairwise CCR value less than that of Gold sequences and large set Kasami sequences. The number of such segments is less than the number of Gold sequences and large set Kasami sequences of same length. The linear complexity properties of segments of binary sequences drived from sequence over Z 4 defined for all six chaotic functions given by Equations (7.1) to (7.6) and using polynomial mapping P 2 (y) are considered in the next section. 7.7 LINEAR COMPLEXITY PROPERTIES OF SEGMENTS OF BINARY SEQUENCES OF LENGTH 15 DERIVED FROM SEQUENCE OVER Z 4 USING POLYNOMIAL MAPPING The binary sequences of length 15 are obtained using proposed method of deriving binary sequences from sequence over Z 4 defined by all the six chaotic map equations and using polynomial mapping P 2 (y) given by Equation (7.8). Based on the magnitude of pairwise CCR value α, segments of sequences are selected for three values of α. 0.6 corresponding to peak magnitude of CCR value of Gold sequences and large set Kasami sequences and two smaller values for which there are no Gold sequences or large set Kasami sequences. These segments of sequences of length 15 are investigated for LC properties. The segments of length 15, having 0.6 which are selected are as listed in Tables 7.59 to 7.64 for all the six chaotic functions defined by Equations (7.1) to (7.6), LC of these sequences are computed. LC is also computed for segments of sequences of same length s, for s = 31, 63, 127 and 255 and it is observed by computer

195 137 search that LC is greater than that of Gold sequences and large set Kasami sequences of same length. Study of LC properties of segments of binary sequences derived from sequence over Z 4 defined by Logistic map Equations and using polynomial mapping P 2 (y)is considered in the next section LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Using Polynomial Mapping Logistic map given by Equation (7.1) is considered first with initial value x 0 = 0.4, r = 3.99 and n = 5. Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 7.59 is considered and their LC is computed. The number of segments out of 100 having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as tabulated in Table 7.59 is considered to study the LC properties. For n = 5 out of 100 segments of length 15, a) 11 segments have magnitude of pairwise CCR value The LC of these 11 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 21 segments have magnitude of pairwise CCR value 0.5. The LC of these 21 sequences is computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 31 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 31 sequences is computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11.

196 138 For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 11. Only sequences corresponding to n= 5 are listed. Table 7.66 Linear complexity of 11 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.67 Linear complexity of 21 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

197 139 Table 7.68 Linear complexity of 31 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC properties of segments of sequences generated using Logistic map given by Equation (7.2) is discussed next. Consider segments of binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.2) with initial value x 0 = 0.4, bifurcation parameter r = 1.99 and using polynomial mapping P 2 (y) given by Equation (7.8). The number of segments of sequences out of 100 segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as listed in Table 7.60 is considered to study the linear complexity properties.

198 140 For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 25 segments have magnitude of pairwise CCR value 0.5. The LC of these 25 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 32 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 32 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 12. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 11. Only sequences corresponding to n= 5 are listed. Table 7.69 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

199 141 Table 7.70 Linear complexity of 25 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.71 Linear complexity of 32 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

200 142 Table 7.71 (Continued) LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map Equation and Using Polynomial Mapping The number of segments of sequences of length 15 out of 100 segments derived from sequence over Z 4 defined by Tent map given by Equation (7.3) and using polynomial mapping P 2 (y) having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.61 are considered. For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 22 segments have magnitude of pairwise CCR value 0.5. The LC of these 22 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 34 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 34 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11.

201 143 For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 11. Only sequences corresponding to n= 5 are listed. Table 7.72 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.73 Linear complexity of 22 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

202 144 Table7.74 Linear complexity of 34 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map Equation and Using Polynomial Mapping To study the LC properties, segments of binary sequences derived from sequence over Z 4 defined by Cubic map given by Equation (7.4) are considered with initial value x 0 = 0.4 and using polynomial mapping P 2 (y). As in the earlier cases the number of segments out of 100 segments having magnitude of pairwise CCR value α less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.62 is considered.

203 145 For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 24 segments have magnitude of pairwise CCR value 0.5. The LC of these 24 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 35 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 35 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table 7.77 In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.75 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

204 146 Table 7.76 Linear complexity of 24 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.77 Linear complexity of 35 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

205 147 Table 7.77 (Continued) LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Using Polynomial Mapping Binary sequences are derived from sequence over Z 4 derived from sequence over Z 4 defined by Quadratic map given by Equation (7.5) is considered next with initial value x 0 = 0.4, r = 0.4 and using polynomial mapping P 2 (y). For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 19 segments have magnitude of pairwise CCR value 0.5. The LC of these 19 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table 7.79.

206 148 c) 27 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 27 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.78 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.79 Linear complexity of 19 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

207 149 Table 7.80 Linear complexity of 27 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Using Polynomial Mapping Bernoulli map given by Equation (7.6) is considered with initial value x 0 = 0.4, r = 1.99 to derive sequence over Z 4. Binary sequence is derived from sequence over Z 4 by using polynomial mapping P 2 (y). The number of segments of length 15 out of 100 segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as summarized in Table 7.64 is considered and their LC properties are investigated.

208 150 For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 21 segments have magnitude of pairwise CCR value 0.5. The LC of these 21 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 28 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 28 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.81 Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

209 151 Table 7.82 Linear complexity of 21 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.83 Linear complexity of 28 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 2 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

210 152 Table 7.83 (Continued) Binary sequences derived from sequence over Z 4 defined by all six chaotic functions given by Equations (7.1) to (7.6) and using polynomial mapping P 2 (y), it is possible to get some segments of sequences of length 15 having LC greater than that of Gold sequences of same length. Having discussed the CCR and LC properties of binary sequences derived from sequences over Z 4 defined by all six chaotic functions and using polynomial mappings P 1 (y)and P 2 (y), CCR properties of segments of binary sequences obtained using polynomial mapping P 3 (y)given by Equation (7.9) is considered in the next section. 7.8 CCR PROPERTIES OF CHAOTIC BINARY SEQUENCES DERIVED FROM SEQUENCE OVER Z 4 USING POLYNOMIAL MAPPING For each of the six chaotic functions defined by Equations (7.1) to (7.6), real valued sequence {x k } and discrete sequence {y k } over Z 4 is generated by choosing the initial value x 0 = 0.4 and multiplication factor n = 5, 6, 7, 8, 9 and 10 as discussed in Section 5.1. This discrete sequence {y k } is converted to binary using polynomial mapping P 3 (y) given by Equation (7.9),. To study the properties of the generated binary sequence, non-overlapping segments of same length s,

211 153 for s = 15, 31, 63, 127 and 255 are considered. The number of segments of different lengths considered in this work is listed in Table 1.1. The number of segments of same length s, for s = 15, 31, 63, 127 and 255 having different magnitude of pairwise CCR values is determined and tabulated in Tables 7.84 to Table 7.84 gives the number of non-overlapping segments of different lengths derived from sequene over Z 4 defined by Logistic map given by Equation (7.1) and using polynomial mapping P 3 (y). It is seen that the number of segments of length 15 is more than number of Gold sequences of same length, having magnitude of pairwsise CCR value 0.6, corresponding to that of Gold sequences of same length.. In this case also there are some segments of sequences less than the number of Gold and large set Kasami sequences of same length, with magnitude of pairwise CCR value α less than that of Gold and large set Kasami sequences. Almost similar results are seen for other chaotic functions defined by Equations (7.2) to (7.6) as tabulated in Tables 7.85 to Table 7.84 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.1) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

212 154 Table 7.85 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Logistic Map Equation (7.2) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.86 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Tent Map Equation (7.3) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.87 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Cubic Map Equation (7.4) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length

213 155 Table 7.88 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Quadratic Map Equation (7.5) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Table 7.89 Number of Segments of Different Lengths of Sequences Derived from Sequence over Z 4 Defined by Bernoulli Map Equation (7.6) and Using Polynomial Mapping P 3 (y) having Different Magnitude of Pairwise CCR Value Trial No. n value For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length Summary of Tables 7.84, 7.85, 7.86, 7.87, 7.88, and 7.89 is given in Table 7.90 which gives the details of chaotic map and maximum number of segments of sequences of same length s, for s = 15, 31, 63, 127 and 255 and whose magnitude of pairwise CCR value is less than three particular values chosen.

214 156 Table 7.90 Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Using Polynomial Mapping P 3 (y) Length of Sequence Value of n Chaotic Map Equation and Table Number Maximum Number of Sequences 0.6 5, 6 Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Tent Map Equation (7.3), Table Cubic Map Equation (7.4), Table Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Cubic Map Equation (7.4), Table Logistic Map Equation (7.1), Table Logistic Map Equation (7.1), Table Bernoulli Map Equation (7.6), Table Cubic Map Equation (7.4), Table Cubic Map Equation (7.4), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.1), Table Bernoulli Map Equation (7.6), Table Logistic Map Equation (7.1), Table Quadratic Map Equation (7.5), Table Bernoulli Map Equation (7.6), Table Quadratic Map Equation (7.5), Table Logistic Map Equation (7.1), Table From the trials carried out and values listed in Tables (7.84) to (7.89), it is seen that for length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 36 with n = 5 or 6 using Quadratic Map Equation (7.5) as given in Table 7.88 and also with n = 10 using Bernoulli map as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 25 atleast for one value of n in all cases of chaotic maps except Logistic map Equations (7.1) and (7.2). iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 13 for n = 7 and Cubic Map Equation (7.4) as given in Table For length 31, i) Maximum number of segments having same CCR as Gold is equal to and is 58 for n = 7 and Logistic Map Equation (7.1) as given in Table 7.84.

215 157 ii) Maximum number of segments having magnitude of pairwise CCR value 0.23 is equal to 28 for n = 7 and Logistic Map Equation (7.1) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 15 for n = 5 and Bernoulli map Equation (7.6) as given in Table For length 63, i) Maximum number of segments having same CCR as Gold is equal to and is 131 for n = 5 and Cubic Map Equation (7.4) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 52 for n = 7 in Cubic map Equation (7.4) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 29 for n = 6 and Bernoulli map Equation (7.6) as given in Table For length 127, i) Maximum number of segments having same CCR as Gold is equal to and is 248 for n = 6 and Logistic Map Equation (7.1) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 89 for n = 7 and Bernoulli map Equation (7.6) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 49 for n = 5 and Logistic Map Equation (7.1) as given in Table For length 255, i) Maximum number of segments having same CCR as Gold is equal to and is 428 with n = 8 using Quadratic Map Equation (7.5) as given in Table 7.88 and also with n = 9 using Bernoulli map Equation (7.6) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value is equal to 141 for n = 8 and Quadratic Map Equation (7.5) as given in Table 7.88.

216 158 iii) Maximum number of segments having magnitude of pairwise CCR value is equal to 89 for n = 10 and Logistic Map Equation (7.1) as given in Table The linear complexity properties of segments of binary sequences drived from sequence over Z 4 defined for all six chaotic functions given by Equations (7.1) to (7.6) and using polynomial mapping P 3 (y)are considered in the next section. 7.9 LINEAR COMPLEXITY PROPERTIES OF SEGMENTS OF BINARY SEQUENCES OF LENGTH 15 DERIVED FROM SEQUENCE OVER Z 4 USING POLYNOMIAL MAPPING The binary sequences are obtained using proposed method of deriving binary sequences from sequence over Z 4 defined by all the six chaotic map equations and using polynomial mapping P 3 (y) given by Equation (7.9). Based on the magnitude of pairwise CCR value α, segments of sequences are selected for three values of α, 0.6 corresponding to peak magnitude of CCR value of Gold sequences and large set Kasami sequences and two smaller values for which there are no Gold sequences or large set Kasami sequences. These segments of sequences of length 15 are investigated for LC properties. The segments of length 15, having 0.6 which are selected are as listed in Tables 7.32 to 7.37 for all the six chaotic functions defined by Equations (7.1) to (7.6), LC of these sequences are computed. LC is also computed for segments of sequences of same length s, for s = 31, 63, 127 and 255 and it is observed by computer search that LC is greater than that of Gold sequences and large set Kasami sequences of same length. To study the LC properties, non-overlapping segments of length 15 obtained for all the six chaotic functions listed in Tables 7.84 to 7.89 are considered. First consider binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.1) and using polynomial mapping P 3 (y).

217 LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Logistic Map Equations and Using Polynomial Mapping Here the LC properties of binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.1) is considered with initial value x 0 = 0.4, r = 3.99, n = 5 and using polynomial mapping P 3 (y) given by Equation (7.9). The number of segments out of 100 segments having magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 as tabulated in Table 7.84 is considered. For n = 5 out of 100 segments of length 15, a) 11 segments have magnitude of pairwise CCR value The LC of these 11 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 21 segments have magnitude of pairwise CCR value 0.5. The LC of these 21 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 30 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 30 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed.

218 160 Table 7.91 Linear complexity of 11 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.92 Linear complexity of 21 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

219 161 Table 7.93 Linear complexity of 30 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.1) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Logistic map given by Equation (7.2) is considered next with x 0 = 0.4 and r = 1.99 for deriving sequence over Z 4. Binary sequences are obtained from sequence over Z 4 using polynomial mapping P 3 (y) given by Equation (7.9). The number of segments out of 100 segments having magnitude of pairwise CCR value α less than or equal to 0.35, 0.5 and 0.6 as listed in Table 7.85 is considered to study the linear complexity properties.

220 162 For n = 5 out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 23 segments have magnitude of pairwise CCR value 0.5. The LC of these 23 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 34 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 34 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.94 Linear complexity of 7 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

221 163 Table 7.95 Linear complexity of 23 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.96 Linear complexity of 34 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Logistic Map Equation (7.2) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

222 164 Table 7.96 (Continued) LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Tent Map Equation and Using Polynomial Mapping Tent map given by Equation (7.3) is considered to derive discrete sequence over Z 4 with initial value x 0 = 0.4 and n = 5. Binary sequence is derived from sequence over Z 4 using polynomial mapping P 3 (y) given by Equation (7.9). As in the earlier cases the number of segments out of 100 segments having magnitude of pairwise CCR value 0.35, 0.5 and 0.6 as summarized in Table 7.86 are considered. For n = 5 out of 100 segments of length 15, a) 12 segments have magnitude of pairwise CCR value The LC of these 12 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 23 segments have magnitude of pairwise CCR value 0.5. The LC of these 23 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 34 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 34 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table 7.99.

223 165 In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table 7.97 Linear complexity of 12 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 7.98 Linear complexity of 23 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

224 166 Table 7.98 (Continued) Table7.99 Linear complexity of 34 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Tent Map Equation (7.3) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

225 167 As in the case of binary sequences of length 15 based on Logistic map Equation (7.1) and (7.2), in the case of Tent map also the linear complexity of some segments is greater than that of Gold sequences of same length. In the next section, deriving binary sequence from sequence over Z 4 defined by Cubic map given by Equation (7.4) and using polynomial mapping P 3 (y) is considered LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Cubic Map Equation and Using Polynomial Mapping Deriving binary sequence from sequence over Z 4 defined by Cubic map given by Equation (7.4) is considered next with initial value x 0 = 0.4, n = 5 and using polynomial mapping P 3 (y). The number of segments out of 100 segments having magnitude of pairwise CCR value 0.35, 0.5 and 0.6 as summarized in Table 7.87 is considered. For n = 5 out of 100 segments of length 15, a) 11 segments have magnitude of pairwise CCR value The LC of these 11 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 23 segments have magnitude of pairwise CCR value 0.5. The LC of these 23 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 26 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 26 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed.

226 168 Table Linear complexity of 11 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table Linear complexity of 23 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

227 169 Table Linear complexity of 26 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Cubic Map Equation (7.4) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Quadratic Map Equation and Using Polynomial Mapping Chaotic binary sequences derived from sequence over Z 4 defined by Quadratic map given by Equation (7.5) is considered next with x 0 = 0.4 and r = 0.4 and using polynomial mapping P 3 (y).

228 170 For n = 5 out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 25 segments have magnitude of pairwise CCR value 0.5. The LC of these 25 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 36 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 36 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table Linear complexity of 7 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

229 171 Table Linear complexity of 25 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table Linear complexity of 36 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Quadratic Map Equation (7.5) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

230 172 Table (Continued) LC of Segments of Binary Sequence Derived from Chaotic Discrete Sequence Defined by Bernoulli Map Equation and Using Polynomial Mapping Bernoulli map given by Equation (7.6) is considered with initial value x 0 = 0.4, r = 1.99 to derive chaotic discrete sequences over Z 4. The sequence over Z4 is mapped to binary using polynomial mapping P 3 (y) given by Equation (7.9). The number of segments of sequences out of 100 segments having magnitude of pairwise CCR value 0.35, 0.5 and 0.6 as summarized in Table 7.85 is considered and their LC properties are studied. For n = 5 out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 25 segments have magnitude of pairwise CCR value 0.5. The LC of these 25 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table

231 173 c) 33 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 33 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. For the 15 bit segments of sequences corresponding to n = 6, 7, 8, 9 and 10, the LC is computed and in these cases also the LC values lies between 8 and 10. Only sequences corresponding to n= 5 are listed. Table Linear complexity of 8 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table Linear complexity of 25 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

232 174 Table (Continued) Table Linear complexity of 33 Binary Sequences of Length 15 Derived from Sequence Over Z 4 Defined by Bernoulli Map Equation (7.6) and Polynomial Mapping P 3 (y) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

233 175 Table (Continued) The linear complexity of binary sequences derived from sequence over Z 4 and polynomial mapping P 3 (y) defined by Equation (7.9) exhibit similar behaviour as that of polynomial mappings P 1 (y) and P 2 (y) defined by equations (7.7) and (7.8) in all the six types of chaotic maps considered. Also it is found that some segments of sequences whose length is 15 have LC greater than that of Gold sequences of same length. It is seen that whatever may be the chaotic map which is used to derive binary sequence from sequence over Z 4 using the four methods of mapping P 0, P 1, P 2 and P 3, their CCR and LC properties are almost identical in all the cases. The number of sequences of any length having pairwise CCR value α less than or equal to that of Gold sequences is always more than the number of Gold sequences of same length. It is also seen that there are set of segments of sequences having pairwise CCR value α less than that of Gold sequences or large set Kasami sequences. Some sequences have LC greater than that of Gold sequences RESULTS AND DISCUSSION It is seen that HCCR property exhibited by discrete sequence over Z 4 obtained using different chaotic maps except Quadratic map, are almost identical for values of n, 5 < n < 10.

234 176 Excellent balance property is exhibited by discrete sequence over Z 4 generated using Tent map and Bernoulli map for the chosen values of n = 5 and x 0 = 0.4 having difference of maximum number of occurrences and minimum number of occurrences equal to 200 and 100 respectively. Table Choice of n and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Z 4 Length of Sequence Value of n Mapping to Binary Using Chaotic Map Equation and Table Number Maximum Number of Sequences 0.6 6, 8 P 1 (y) Logistic Map Equation (7.2), Table P 1 (y) Bernoulli Map Equation (7.6), Table P 2 (y) Tent Map Equation (7.3), Table P 2 (y) Quadratic Map Equation (7.5), Table P 3 (y) Cubic Map Equation (7.4), Table P 2 (y) Logistic Map Equation (7.2), Table P 1 (y) Logistic Map Equation (7.2), Table P 2 (y) Logistic Map Equation (7.2), Table P 1 (y) Logistic Map Equation (7.2), Table P 1 (y) Tent Map Equation (7.3), Table P 1 (y) Bernoulli Map Equation (7.6), Table P 1 (y) Logistic Map Equation (7.1), Table P 3 (y) Cubic Map Equation (7.4), Table P 1 (y) Logistic Map Equation (7.1), Table P 1 (y) Logistic Map Equation (7.2), Table P 1 (y) Cubic Map Equation (7.4), Table P 2 (y) Bernoulli Map Equation (7.6), Table P 2 (y) Logistic Map Equation (7.2), Table P 2 (y) Quadratic Map Equation (7.5), Table P 3 (y) Bernoulli Map Equation (7.6), Table P 2 (y) Logistic Map Equation (7.1), Table P 2 (y) Cubic Map Equation (7.4), Table P 3 (y) Logistic Map Equation (7.1), Table P 2 (y) Bernoulli Map Equation (7.6), Table P 2 (y) Tent Map Equation (7.3), Table P 2 (y) Logistic Map Equation (7.1), Table P 3 (y) Logistic Map Equation (7.1), Table 7.84 Table summarizes Tables 7.15, 7.40, 7.65 and It gives the choice of chaotic map with x 0 = 0.4 and binary map which gives maximum number of segments of binary sequences derived from sequence over Z 4 and of different length s, having peak magnitude of pairwise CCR value less than or equal to three chosen values of α. First column in Table gives the length of the sequence, second column gives magnitude of pairwise CCR values chosen, third column gives values of n, fourth column gives

235 177 binary mapping, fifth column lists the type of chaotic map and corresponding Table number from which the maximum number is obtained. Last column gives the maximum number of sequences. From Table 7.109, it is seen that for segment of length 15, i) Maximum number of segments having same CCR as Gold sequence is equal to 41 with n = 6 or 8 using Logistic Map Equation (7.2) and using polynomial mapping P 1 (y) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 26 with n = 8 using Bernoulli Map Equation (7.6) and using polynomial mapping P 1 (y) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 13 using Tent map or Quadratic map with n =6 and polynomial mapping P 2 (y) or with n = 7 using Cubic map and polynomial mapping P 3 (y). Similarly for length 31 from Table 8.55, the maximum number of segments having same CCR as Gold sequence is equal to 60 for n = 6 using Logistic Map Equation (7.2) and polynomial mapping P 2 (y) as given in Table For length 63, maximum number of segments having same CCR as Gold sequence is equal to and is 138 for n = 7 using Logistic Map Equation (7.1) and polynomial mapping P 1 (y) as given in Table For length 127, maximum number of segments having same CCR as Gold sequence is equal to and is 256 for n = 6 using Bernoulli Map Equation (7.6) and polynomial mapping P 2 (y) as given in Table For length 255, maximum number of segments having same CCR as Gold sequence is equal to and is 488 for n = 5 using Bernoulli Map Equation (7.6) and polynomial mapping P 2 (y) as given in Table Similarly it is shown for other cases.

236 178 It is also seen that the LC of some segments of binary sequences of length 15 is equal to 12 with n = 5 in case of binary sequences derived from sequence over Z 4 defined by Logistic map given by Equation (7.2) and using polynomial mapping P 1 (y) or using Tent map given by Equation (7.3) and polynomial mapping P 3 (y) given by Equation (7.9). The period of the proposed binary sequences is found to be large (more than ) for all the six chaotic functions considered and for choice of initial value x 0 equal to 0.4, choice of values of n equal to 5, 6, 7, 8, 9 and 10 with m equal to 4. However, a long period is desirable for cryptographic applications and for CDMA applications required length is much less than the period of the sequence. The chaotic binary sequences derived using six chaotic map equations, have almost same pairwise CCR value, mean square cross correlation value and almost same number of sequences of given length. The pairwise CCR value and mean square CCR values are less compared to the binary sequences derived using earlier methods like using threshold function and using quantization and encoding of CML. As a result, use of proposed binary sequences give rise to small average BER for given number of users and given compared to earlier methods as discussed in Section 1.7. Table Comparison of peak magnitude of Pairwise CCR value of Proposed Sequences with Earlier Methods Sl. No N Chaotic Function Binary Sequence Derived using Logistic Map, Tent Map, Bernoulli Map and Chebyshev Map Logistic Map, Tent Map Logistic Map Threshold Function Threshold Function Quantization & Encoding of CML Reference Hongtao Zhang et al (2000) and (2001) Soobul Y. et al (2002) Hu Saigui (1996) CCR Value Proposed Binary Sequences CCR Value

237 179 Table gives a comparison of proposed binary sequences derived from chaotic sequences with the earlier methods using threshold function and using quantization and encoding of CML. It is found that the proposed segments of sequences have low value of peak magnitude of pairwise CCR value compared to earlier methods as discussed in some papers. Bound on CCR value of proposed sequences is discussed in Chapter SUMMARY Method of generation of chaotic discrete sequence over Z 4 and chaotic binary sequences using chaotic map equations proposed in Chapter 5 is considered in this chapter and their CCR and LC properties are investigated. The generated discrete sequences are analyzed for Hamming cross correlation and balance properties. The segments of discrete sequences are found to have low values of cross correlation and good balance properties. The chaotic binary sequences derived from sequence over Z 4 using mapping P 0 and three polynomial mappings P 1, P 2, P 3 also exhibit desirable CCR and LC properties compared to Gold sequences and large set Kasami sequences of same length. It is found that the period of chaotic binary sequences generated using proposed scheme is much larger than the length of the segment of binary sequences. The pairwise CCR value is lower than binary sequences derived using earlier methods like using threshold function and using quantization and encoding of CML. Udaya Siddiqi sequences satisfy Welch bound and Sidelnikov bound with equality. The number of such sequences is equal to (2 r ), where r is an integer. The length of the sequence is 2 (2 r - 1). Using proposed method of deriving chaotic binary sequences from chaotic discrete sequences, there are inifinite number of choices for initial values, number of choices for n and number of choices for m.

238 CHAPTER 8 PROPERTIES OF CHAOTIC DISCRETE SEQUENCES OVER GF(2 m ) AND CORRESPONDING BINARY SEQUENCES DERIVED FROM THEM

239 180 CHAPTER 8 PROPERTIES OF CHAOTIC DISCRETE OVER GF(2 m ) AND CORRESPONDING BINARY SEQUENCES DERIVED FROM THEM The generation of chaotic discrete sequences over finite field of order 2 m defined by two Logistic maps and methods of obtaining binary sequences from this are discussed in Section 6.2 and 6.3. As discussed in Chapter 6 only two Logistic maps over GF(2 m ) are considered, which generates discrete sequences of appreciable period over GF(2 m ). As discussed in Section 6.1, Logisitc map given by Equation (6.1) can be regarded as first order NLRR and variation in Logistic map given by Equation (6.2) can be regarded as first order inhomogeneous LRR. The periodicity, cross correlation and linear complexity properties of the chaotic sequences generated using Equations (6.1) and (6.2) are different from sequences generated using first order homogeneous LRR. Properties of sequences generated using homogeneous LRR over finite field are well established (Rudolf Lidl and Herald Niederriter 1984). In the next section, normalized cyclic Hamming Autocorrelation (HACR) and balance properties of discrete chaotic sequences over GF(2 m ) is considered. The pairwise CCR properties and LC of segments of binary sequences are investigated later.

240 HAMMING AUTOCORRELATION (HACR) PROPERTIES AND BALANCE PROPERTIES OF SEQUENCES OVER GF(2 m ) Two chaotic map equations over GF(2 m ) are proposed and are based on Logistic map Equation (6.1) and (6.2) and are repeated here. (8.1) (8.2) The addition and multiplication operation is defined in GF(2 m ) with reference to an irreducible polynomial of degree m over GF(2). Chaotic sequences over finite field GF(2 m ) are of finite length. As shown in Note 6.1 and 6.2, certain choice of initial values results in generation of trivial sequences of period one. Theorem 8.1 stated below gives a bound on the period of sequences over GF(2 m ). Theorem 8.1: Consider one periodic length M of sequence over GF(2 m ). Then M is always less than 2 m. Proof: From the result of Note 6.1 and Note 6.2 some initial values produce trivial sequence of period 1. Hence all the elements as initial value in GF(2 m ) will not generate sequence of period greater than 1. Any sequence of period M will not contain elements which give trivial sequence. Hence in the sequence of one period over GF(2 m ), the number of elements is less than 2 m. Hence M is less than 2 m. Theorem 8.2: The normalized cyclic HACR function of discrete sequence of period M > 1, over GF(2 m ) has two values + 1 for shift τ = 0 and -1 for τ 0. Proof: In one periodic length M of sequence any element occurs only once. Hence from the definition of normalized HACR function (Equation (4.1.10)) for τ = 0, there is agreement in all the M locations and HACR value is 1. For τ 0, since every element in the sequence occurs only once, there is disagreement in all the M locations. Hence HACR value for τ 0 is 1.

241 182 In this work m = 8 and m = 16 are considered. The Theorem holds good for both these cases. HACR properties of sequences over GF(2 8 ) are considered in the next section HACR Properties of Sequences over GF(2 8 ) As discussed in Chapter 6, using Logistic map Equation (8.1) and nine initial values x 0 = g, g 2.g 9 and 24 values of each r 1 and r 2 equal to g, g 2.g 24, discrete chaotic sequences are generated over GF(2 8 ) defined by irreducible polynomial (x 8 +x 6 +x 5 +x 4 +1), using computer simulation. It is found that maximum possible period is 63. Likewise using Logistic map Equaion (8.2), it is found that the maximum possible period of discrete chaotic sequence over GF(2 8 ) is 255. The sequences generated using Equations (8.1) and (8.2) are investigated for HACR and balance properties. If sequences of one periodic length say 63 or 255 over GF(2 8 ) is considered, then any element in the sequence occurs only once. Hence the sequences have two level normalized cyclic HACR. From Theorem 8.2 normalized HACR is +1 for τ = 0 and -1 for τ 0. Also from Theorem 8.1, the period N of sequences over GF(2 8 ) is less than Balance Properties Since the period of sequence generated over GF(2 8 ) using Equation (8.1) is only 63, which is less than maximum period 256, all the elements are not present in the sequence over GF(2 8 ). But every element in a sequence of period 63 occurs only once. The balance property is satisfied in case of chaotic sequences over GF(2 8 ) generated using Equation (8.2), since the maximum possible period is 255. Here except one element, all the elements occur only once.

242 183 In the next section, the properties of chaotic binary sequences derived from chaotic sequence over GF(2 8 ) is considered. 8.2 CCR PROPERTIES OF SEGMENTS OF BINARY SEQUENCES DERIVED FROM CHAOTIC SEQUENCES OVER GF(2 8 ) Here GF(2 8 ) is defined by modulo a primitive polynomial (x 8 +x 6 +x 5 +x 4 +1) over GF(2). The random sequence of finite field elements is generated using chaotic functions defined by Logistic maps given by Equations (8.1) and (8.2). The initial values x 0, bifurcation parameters r 1 and r 2 are all from finite field GF(2 8 ) of order 256. Hence the total number of possible combinations of x 0, r 1 and r 2 is 256 x 256 x 256. By computer simulation for arbitrarily chosen 9 initial values of x 0 equal to g, g 2.g 9, 24 values of each r 1 and r 2 equal to g, g 2.g 24 sequences over GF(2 8 ) are generated. This leads to 9 x 24 x 24 = 5184 choices of x 0, r 1 and r 2 out of (256) 3 all possible choices. For these 5184 combinations the period of chaotic sequence over GF(2 8 ) is found to be less than or equal to 63 using Logistic map given by Equation (8.1) and less than or equal to 255 using Equation (8.2). Because of smaller period in GF(2 8 ) the number of segments of sequences are less. Hence the segments of sequences of length 15 are only considered to study the CCR and LC properties in case of GF(2 8 ). Number of segments of sequences of length 15 available in case of GF(2 16 ) is more than that of GF(2 8 ). The chaotic sequences defined over GF(2 16 ) have longer period. Hence segments of binary sequences of length not only 15 but greater than 15 are also considered. As in earlier cases discussed, s is chosen to be 15, 31, 63, 127 and 255 for studying CCR and LC properties and are discussed in Section 8.5.

243 184 The binary sequences are derived from sequence over GF(2 8 ) using the three methods as discussed in Section 6.3.1, and namely, i) Expressing every element in GF(2 8 ) of the sequence as binary 8 tuple ii) Selecting a particular binary bit from each element of the sequence over GF(2 8 ) iii) Mapping every element in GF(2 8 ) to GF(2) using Trace function are considered for deriving binary sequences from sequence over GF(2 8 ) for Equations (8.1) and (8.2). The pairwise CCR properties are investigated for the segments of binary sequences of length 15 derived from sequence over GF(2 8 ). In this case also three values of α are considered. In the next section, pairwise CCR properties of binary sequences generated by expressing field element in GF(2 8 ) as binary 8 tuple is considered CCR Properties of Binary Sequences Generated by Expressing Field Element in GF(2 8 ) as Binary 8 Tuple As discussed in Section 6.3.1, discrete chaotic sequence over GF(2 8 ) with period 63 using Logistic map Equation (8.1) and sequences with period 255 using Logistic map Equation (8.2) are selected by computer simulation. The elements in sequence are represented by binary 8 tuples. Binary sequence is obtained by concatenating these 8 tuple. The resulting binary sequences have length 504 or To study the CCR properties, 33 non-overlapping segments of length 15 are considered as discussed in Section The 15 bit segments are numbered as Segment1, Segment2 Segment33 as shown in Figure 1.1. The number of segments of length 15 selected whose magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6. The results obtained for Logistic map Equation (8.1) and for different α are summarized in Table 8.1 for binary sequences derived by expressing filed element in GF(2 8 ) as binary 8 tuple for different values of x 0, r 1 and r 2. From Table 109, for x 0 = g, r 1 = g 23 and r 2 = g 24 the number of segments of sequences having magnitude of pairwise CCR value 0.35 is found to be 4. Similarly for magnitude of pairwise CCR value

244 , there are 8 segments of sequences and for magnitude of pairwise CCR value 0.6, number of segments of sequences is 19. It is seen that there are more number of segments of sequences compared to the number of Gold sequences of same length having magnitude of pairwise CCR value 0.6. There are some segments of sequences whose magnitude of pairwise CCR value less than 0.6, corresponding to Gold sequences. Table 8.1 Number of Segments out of 33 Segments Derived from Sequence over GF(2 8 ) defined by Logistic map Equation (8.1) and by Expressing Element in GF(2 8 ) as 8 tuple having Magnitude of Pairwise CCR Value 0.35, 0.5 and 0.6 Trial Number x 0 chosen r 1 chosen r 2 chosen Number of binary sequences having magnitude of pairwise CCR value α g g 23 g g 2 g 23 g g 3 g 23 g g 4 g 23 g g 5 g 23 g g 6 g 23 g g 7 g 23 g g 8 g 23 g g 9 g 23 g g g 24 g g 2 g 24 g g 3 g 24 g g 4 g 24 g g 5 g 24 g g 6 g 24 g g 7 g 24 g g 8 g 24 g g 9 g 24 g It is observed that by modifying Logistic map Equation (8.1) into the form given in Equation (8.2), it is possible to obtain period of 255 elements over GF(2 8 ). Each of these elements are expressed in binary 8 tuple and concatenated to get binary sequence of length 2040 and results in 136 non-overlapping segments of 15 as discussed in Section The results are tabulated in Table 8.2 with r 2 = g and different values of x 0 and r 1. It is seen from Table 8.2 that number of segments of sequences are almost double that of Gold sequences of length 15 having pairwise 0.6. There are also some segments of sequences with α < 0.6.

245 186 Table 8.2 Number of Segments out of 136 Segments Derived from Sequence over GF(2 8 ) defined by Logistic map Equation (8.2) and by Expressing Element in GF(2 8 ) as 8 tuple having Magnitude of Pairwise CCR Value 0.35, 0.5 and 0.6 Trial Number x 0 chosen r 1 chosen Number of binary sequences having magnitude of pairwise CCR value CCR value α g g g 2 g g 3 g g 4 g g 5 g g 6 g g 7 g g 8 g g 9 g g g g 2 g g 3 g g 4 g g 5 g g 6 g g 7 g g 8 g g 9 g Table 8.3 lists the summary of Tables 8.1 to 8.2. The segments of length s = 15 and with initial value x 0 = 0.4 is considered. First column in Table 8.3 column gives magnitude of pairwise CCR value, second column gives value of x 0, third column gives value of r 1, fourth column gives value of r 2, fourth column gives chaotic map and corresponding Table number from which maximum number of segments of sequences are considered which is given in the last column.

246 187 Table 8.3 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 8 ) by Expressing Element in GF(2 8 ) as 8 tuple Value of x 0 Value of r 1 Value of r 2 Chaotic Map Equation and Table Number Maximum Number of Sequences of length g 2 g 2 g Logistic Map Equation (8.2), Table g 2 g 3 g 4 g 7 g 9 g 9 g 23 g g 24 g Logistic Map Equation (8.1), Table 8.1 Logistic Map Equation (8.2), Table g 7 g 2 g Logistic Map Equation (8.2), Table From the trials carried out and values listed in Tables 8.1 and 8.2, it is seen that, for segments of length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 38 with x 0 = g 2, r 1 = g 2 and r 2 = g using Logistic Map Equation (8.2) as given in Table 8.2. ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 10 with x 0 = g 2 or g 3 or g 4 or g 7 or g 9, r 1 = g 2 and r 2 = g using Logistic Map Equation (8.1) as given in Table 8.1 and also with x 0 = g 9, r 1 = g and r 2 = g using Logistic Map Equation (8.2) as given in Table 8.2. iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 5 with x 0 = g 7, r 1 = g 2 and r 2 = g using Logistic Map Equation (8.2) as given in Table 8.2. In the next section, CCR properties of binary sequences generated by selecting a particular bit from each element of the sequence over GF(2 8 ) is considered.

247 CCR Properties of Binary Sequences Generated by Selecting a Particular Bit From Each Element of the Sequence Over GF(2 8 ) As discussed in Section 6.3.2, discrete chaotic sequence over GF(2 8 ) with period 63 using Logistic map Equation (8.1) and with period 255 using Logistic map Equation (8.2) are selected by computer simulation. The elements in sequence are represented by binary 8 tuples. Binary sequence is obtained by selecting a particular bit from each element and concatenating the corresponding bits. Hence choosing bits in location i (i = 1 to 8) results in 8 sequences of length 63 using Logistic map given by Equation (8.1). For each of these sequences there are only 4 non-overlapping segments of length 15. The results are obtained for 4 segments of length 15 with magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6 and are tabulated in Table 8.4. From Table 8.4, it is seen that number of sequences having magnitude of pairwise CCR value 0.6 is 4 and there are some segments of sequences with α < 0.6. Table 8.4 Number of Segments out of 4 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 8 ) of Length 15 having Magnitude of Pairwise CCR Value 0.35, 0.5 and 0.6 Bit Chosen i Number of binary sequences having magnitude of pairwise CCR value α (MSB) (LSB) 3 4 4

248 189 Next using Logistic map Equation (8.2), sequence of length 255 elements over GF(2 8 ) is generated. By selecting a particular bit in location i (i = 1 to 8) results in 8 sequences of length 255 which is divided into 17 non-overlapping segments of length 15. The number of segments selected for three different values of α are listed in Table 8.5. From Table 8.5 it is seen that the number of segmens of sequences having pairwise 0.6 is about 15 and there are some segments of sequences with α < 0.6. Table 8.5 Number of Segments out of 17 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Selecting a Particular Bit from Element Over GF(2 8 ) of Length 15 having Magnitude of Pairwise CCR Value 0.35, 0.5 and 0.6 Bit Chosen i Number of binary sequences having magnitude of pairwise CCR value α (MSB) (LSB) Table 8.6 lists the summary of Tables 8.4 to 8.5 and gives maximum number of sequences of length 15 with magnitude of pairwise CCR value less than or equal to given α. Table 8.6 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 8 ) by Selecting a Particular Bit from Element Over GF(2 8 ) Value of x 0 Value of r 1 Value of r 2 Chaotic Map Equation and Table Number Maximum Number of Sequences of length g 2 g 2 g Logistic Map Equation (8.2), Table g 2 g 23 g 24 Logistic Map Equation (8.2), Table g 7 g 2 g Logistic Map Equation (8.2), Table 8.5 4

249 190 From the trials carried out and values listed in Tables 8.4 and 8.5, it is seen that, for segments of length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 16 with x 0 = g 2, r 1 = g 2 and r 2 = g using Logistic Map Equation (8.2) as given in Table 8.5. ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 11 with x 0 = g 2, r 1 = g 23 and r 2 = g 24 using Logistic Map Equation (8.2) as given in Table 8.5. iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 4 with x 0 = g 7, r 1 = g 2 and r 2 = g using Logistic Map Equation (8.2) as given in Table 8.5. In the next section, CCR properties of binary sequences generated by mapping every element in GF(2 8 ) to GF(2) using Trace function is discussed CCR Properties of Binary Sequences Generated by Mapping Every Field Element in GF(2 8 ) to GF(2) Using Trace Function As discussed in Section using Trace function, every 8 tuple in GF(2 8 ) is mapped to distinct element in GF(2). The random chaotic sequence over GF(2 8 ) generated using Logistic map given by Equation (8.1) has a period 63 and when transformed to binary using trace function also has the period 63. This results in 4 nonoverlapping segments of length 15. The results are obtained for these 4 segments of length 15 with magnitude of pairwise CCR value less than or equal to 0.35, 0.5 and 0.6. The results obtained for three values of α are summarized in Table 8.7 for different x 0, r 1 and r 2.

250 191 Table 8.7 Number of Segments out of 4 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and Using Trace Function having Magnitude of Pairwise CCR Value 0.35, 0.5 and 0.6 Trial Number x 0 chosen r 1 chosen r 2 chosen Number of binary sequences having magnitude of pairwise CCR value α g g 23 g g 2 g 23 g g 3 g 23 g g 4 g 23 g g 5 g 23 g g 6 g 23 g g 7 g 23 g g 8 g 23 g g 9 g 23 g g g 24 g g 2 g 24 g g 3 g 24 g g 4 g 24 g g 5 g 24 g g 6 g 24 g g 7 g 24 g g 8 g 24 g g 9 g 24 g Using Equation (8.2), the maximum possible period is 255 for chaotic sequence over GF(2 8 ) and when transformed to binary using trace function also results in the same period of 255. The results for three values of α are tabulated in Table 8.8 with r 1 = g, for different x 0 and r 1. In this case also it is seen that number of sequences having magnitude of pairwise CCR value 0.6 is 10 to 14 depending on the choice of x 0, r 1 and r 2 and there are some segments of sequences with α < 0.6.

251 192 Table 8.8 Number of Segments out of 17 Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and Using Trace Function having Magnitude of Pairwise CCR Value 0.35, 0.5 and 0.6 Trial Number x 0 chosen r 2 chosen Number of binary sequences having magnitude of pairwise CCR value CCR value α g g g 2 g g 3 g g 4 g g 5 g g 6 g g 7 g g 8 g g 9 g g g g 2 g g 3 g g 4 g g 5 g g 6 g g 7 g Table 8.9 lists the summary of Tables 8.7 to 8.8 and gives maximum number of sequences of length 15 with magnitude of pairwise CCR value less than or equal to given α. Table 8.9 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 8 ) using Trace Function Value of x 0 Value of r 1 Value of r 2 Chaotic Map Equation and Table Number Maximum Number of Sequences of length g 5 g g 2 Logistic Map Equation (8.2), Table g 5 g g 2 Logistic Map Equation (8.2), Table g 7 g g Logistic Map Equation (8.2), Table From the trials carried out and values listed in Tables 8.7 and 8.8, it is seen that, for segments of length 15,

252 193 i) Maximum number of segments having same CCR as Gold is equal to and is 15 with x 0 = g 5, r 1 = g and r 2 = g 2 using Logistic Map Equation (8.2) as given in Table 8.7. ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 8 with x 0 = g 5, r 1 = g and r 2 = g 2 using Logistic Map Equation (8.2) as given in Table 8.7. iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 5 with x 0 = g 7, r 1 = g and r 2 = g using Logistic Map Equation (8.2) as given in Table 8.7. In Sections , and , it is shown that, there are some segments of sequences of length 15 with magnitude of pairwise CCR value 0.35 or 0.5 or 0.6 corresponding to Gold sequences and large set Kasami sequences of same length. LC properties of binary sequences derived from chaotic sequence over GF(2 8 ) having desirable magnitude of pairwise CCR values of length 15 is considered in the next section. 8.3 LC PROPERTIES OF SEGMENTS OF BINARY SEQUENCES OF LENGTH 15 DERIVED FROM CHAOTIC SEQUENCES OVER GF(2 8 ) Based on the magnitude of pairwise CCR value α, segments of sequences are selected for three values of α. 0.6 corresponding to peak magnitude of CCR value of Gold sequences and large set Kasami sequences and two smaller values for which there are no Gold sequences or large set Kasami sequences. For trial number 1 in Table 8.1 out of 33 segments, 4 segments have magnitude of pairwise CCR value 0.35, 8 segments have magnitude of pairwise CCR value 0.5 and 19 segments have magnitude of pairwise CCR value 0.6. LC properties of these segments are considered in the next section.

253 LC Properties of Binary Sequences of length 15 Generated by Expressing Field Element in GF(2 8 ) as Binary 8 Tuple To study the LC properties of segments of sequences of length 15 exhibiting desirable pairwise CCR values which are derived from Logistic map given by Equation (8.1) and by expressing field element in GF(2 8 ) as binary 8 tuples, Table 8.1 is considered. Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 8.1 is considered and their LC is computed. Out of 33 segments of length 15, a) 4 segments have magnitude of pairwise CCR value The LC of these 4 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 8 segments have magnitude of pairwise CCR value 0.5. The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 19 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 19 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 9. Table 8.10 Linear Complexity of 4 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

254 195 Table 8.11 Linear Complexity of 8 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.12 Linear Complexity of 19 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity LC properties of segments of sequences generated using Logistic map Equation (8.2) is discussed next. The segments of sequences of length 15 exhibiting desirable

255 196 pairwise CCR values which are derived from Logistic map Equation (8.2) and by expressing field element in GF(2 8 ) as binary 8 tuples, Table 8.2 is considered. Out of 136 segments of length 15, a) 2 segments have magnitude of pairwise CCR value The LC of these 2 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 5 segments have magnitude of pairwise CCR value 0.5. The LC of these 5 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 30 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 30 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 11. Table 8.13 Linear Complexity of 2 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.14 Linear Complexity of 5 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

256 197 Table 8.15 Linear Complexity of 30 Binary Segments Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 8 ) as 8 Tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity In the next section, LC properties of non-overlapping segments of length 15 derived form sequence over GF(2 8 ) defined by Logistic map Equations (8.1), (8.2) and by selecting a particular bit from each element of the sequence over GF(2 8 ) is considered.

257 LC Properties of Binary Sequences of length 15 Generated by Selecting a Particular Bit from Each Element of the Sequence Over GF(2 8 ) To study the LC properties of non-overlapping segments of length 15 exhibiting desirable pairwise CCR values which are derived from sequence over GF(2 8 ) defined by Logistic map Equation (8.1) and by selecting the MSB bit from each element of the sequence over GF(2 8 ), Table 8.4 is considered. Out of 4 segments of length 15, a) 2 segments have magnitude of pairwise CCR value The LC of these 2 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 3 segments have magnitude of pairwise CCR value 0.5. The LC of these 3 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 4 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 4 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 9. Table 8.16 Linear Complexity of 2 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

258 199 Table 8.17 Linear Complexity of 3 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.18 Linear Complexity of 4 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity To study the LC properties of non-overlapping segments of length 15 derived from sequence over GF(2 8 ) defined by Logistic map Equation (8.2) and by selecting the MSB bit from element over GF(2 8 ), set of sequences listed in Table 8.5 for trial number 1 is considered. Out of 17 segments of length 15, a) 4 segments have magnitude of pairwise CCR value The LC of these 4 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 11 segments have magnitude of pairwise CCR value 0.5. The LC of these 11 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 15 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 15 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 9.

259 200 Table 8.19 Linear Complexity of 4 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.20 Linear Complexity of 11 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.21 Linear Complexity of 15 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and by Selecting a Particular Bit from Element Over GF(2 8 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

260 201 Table 8.21 (Continued) LC properties of non-overlapping segments of length 15 derived from sequence over GF(2 8 ) defined by Logistic map given by Equations (8.1), (8.2) and by mapping every field element in GF(2 8 ) to GF(2) using trace function is considered in the next section LC Properties of Binary Sequences of Length 15 Generated by Mapping Every Field Element in GF(2 8 ) to GF(2) Using Trace Function To study the LC properties, non-overlapping segments of length 15 exhibiting desirable pairwise CCR values which are derived from sequence over GF(2 8 ) defined by Logistic map Equation (8.1) and by mapping every field element in GF(2 8 ) to GF(2) using trace function, Table 8.7 is considered. Out of 4 segments of length 15, a) 2 segments have magnitude of pairwise CCR value The LC of these 2 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 3 segments have magnitude of pairwise CCR value 0.5. The LC of these 3 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table 8.23.

261 202 c) 3 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 3 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value is 8. Table 8.22 Linear Complexity of 2 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and using Trace Function having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.23 Linear Complexity of 3 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and using Trace Function having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.24 Linear Complexity of 3 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.1) and using Trace Function having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity To study the LC properties, non-overlapping segments of length 15 exhibiting desirable pairwise CCR values which are derived from sequence over GF(2 8 ) defined by Logistic map Equation (8.2) and by mapping every field element in GF(2 8 ) to GF(2) using trace function, Table 8.8 is considered.

262 203 Out of 17 segments of length 15, a) 2 segments have magnitude of pairwise CCR value The LC of these 2 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 6 segments have magnitude of pairwise CCR value 0.5. The LC of these 6 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 12 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 12 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 9. Table 8.25 Linear Complexity of 2 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and using Trace Function having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.26 Linear Complexity of 6 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and using Trace Function having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

263 204 Table 8.27 Linear Complexity of 12 Binary Sequences Derived from Sequence over GF(2 8 ) Defined by Logistic Map Equation (8.2) and using Trace Function having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity In the next section, normalized cyclic Hamming Autocorrelation (HACR) and balance properties of discrete chaotic sequences over GF(2 16 ) which is found to be similar to that of sequences over GF(2 8 ) is considered. The pairwise CCR properties and LC of segments of binary sequences are investigated later. 8.4 HACR PROPERTIES AND BALANCE PROPERTIES OF SEQUENCES OVER GF(2 16 ) As discussed in Chapter 6, using Logistic map Equation (8.1) and nine initial values x 0 = g, g 2.g 9 and 24 values of each r 1 and r 2 equal to g, g 2.g 24, discrete chaotic sequences are generated over GF(2 16 ) defined by irreducible polynomial (x 16 +x 5 +x 3 +x 2 +1), using computer simulation. It is found that maximum possible period is Likewise using Logistic map Equaion (8.2), it is found that the maximum possible period of discrete chaotic sequence over GF(2 16 ) is The generated chaotic sequences over GF(2 16 ) are investigated for HACR properties in the next section followed by the balance property.

264 HACR Properties of Sequences over GF(2 16 ) Considering sequences of one periodic length say or over GF(2 8 ), then any element in the sequence occurs only once. Hence from Theorem 7.2, the sequences have two level normalized cyclic HACR values which will be +1 for τ = 0 and -1 for τ 0. Also from Theorem 7.1, the period M of sequences over GF(2 16 ) is always less than 2 16 = Balance Properties of Sequences over GF(2 16 ) Since the period in case of chaotic sequences over GF(2 16 ) generated using Equation (8.1) is 14322, which is less than maximum period 65535, all the elements are not present in the sequence over GF(2 16 ) and hence the balance property is not good. But every element in the sequence occurs only once. The balance property is satisfied in case of chaotic sequences over GF(2 16 ) generated using Equation (8.2), since the maximum possible period is Here except one element, all the elements occur only once in one period. Properties of chaotic binary sequences derived from chaotic sequence over GF(2 16 ) using three methods as discussed in Section 6.4 is considered in the next section. To study the CCR and LC properties of non- overlapping segments of length s, for s = 15, 31, 63, 127 and 255, chaotic binary sequences are derived from chaotic sequence over GF(2 16 ) with addition multiplication modulo, a primitive polynomial (x 16 +x 5 +x 3 +x 2 +1) over GF(2). The number of segments considered is listed in Table 1.1. In these cases also it is seen that there are some segments of sequences with magnitude of pairwise CCR value less than that of Gold sequences and large set Kasami sequences

265 206 of same length. Properties of segments of binary sequences derived from chaotic sequence over GF(2 16 ) is considered next. 8.5 CCR PROPERTIES OF SEGMENTS OF BINARY SEQUENCES DERIVED FROM CHAOTIC SEQUENCES OVER GF(2 16 ) The random finite field elements generated using Logistic map Equations (8.1) and (8.2) over GF(2 16 ) are mapped to binary sequence using three methods as discussed in Section 7.7. The initial values x 0, bifurcation parameters r 1 and r 2 are all from GF(2 16 ) and the total number of non zero field elements are for each x 0, r 1 and r 2. Hence the total number of possible input combinations is x x To study the cross correlation and linear complexity properties of generated binary sequences, 9 initial values x 0 equal to g, g 2 g 9 and values of each r 1 and r 2 equal to g are considered. The binary sequences are derived from sequence over GF(2 16 ) using the three methods as discussed in Section 6.4.1, and namely, i) Expressing every element in GF(2 16 ) of the sequence as binary 16 tuple ii) Selecting a particular binary bit from each element of the sequence over GF(2 16 ) iii) Mapping every element in GF(2 16 ) to GF(2) using Trace function are considered for deriving binary sequences from sequence over GF(2 16 ) for Equations (8.1) and (8.2). The pairwise CCR properties are investigated for the segments of binary sequences of same length s, for s = 15, 31, 63, 127 and 255 derived from sequence over GF(2 16 ). Details of non-overlapping segments of different lengths considered in this study are listed in Table 1.1. In this case also three values of α is considered and the segments of sequences are selected based on this pairwise α value. In the next section, CCR properties of chaotic binary sequences generated by expressing field element in GF(2 16 ) as binary 16 tuple governed by Logistic map given by Equations (8.1) and (8.2) is considered.

266 CCR Properties of Binary Sequences Generated by Expressing Field Element in GF(2 16 ) as Binary 16 Tuple Discrete chaotic sequence over GF(2 16 ) with period using Logistic map Equation (8.1) and with period using Logistic map Equation (8.2) are selected by computer simulation as discussed in Chapter 6. The elements in sequence are represented by binary 16 tuples and binary sequence is obtained by concatenating these 16 tuple. The resulting binary sequences have length or The period of the binary sequence derived by Expressing Field Element in GF(2 16 ) as Binary 16 tuple defined by Logistic map given by Equations (8.1) as discussed in Section is and defined by Equation (8.2). The binary sequence is then divided into non-overlapping segments of same length s, for s = 15 bit, 31 bit, 63 bit, 127 bit and 255. The number of non-overlapping segments of length 15 using Equation (8.1) is and for Equation (8.2). Only 100 non-overlapping segments of length 15 are considered to study the CCR properties and each of these segments are numbered as Segment 1, Segment 2 as shown in Figure 1.1. Details of non-overlapping segments of different lengths considered in this study are listed in Table 1.1. The CCR properties of the binary sequences derived from sequence over GF(2 16 ) defined by Logistic map Equations (8.1) and (8.2) and by expressing field element in GF(2 16 ) as binary 16 tuple for different α are summarized in Table 8.28 and 8.29 respectively.

267 208 Table 8.28 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 16 ) as 16 Tuple having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g Trial No. x 0 chose n For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length g g g g g g g g g Table 8.29 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 16 ) as 16 Tuple having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g Trial No. x 0 chose n For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length g g g g g g g g g It is seen from Table 8.28 and 8.29 that, the number of segments of sequences are more than the number of Gold sequences of same length. Also there are some segments of binary sequences having magnitude of pairwise CCR value less than that of Gold and large set Kasami sequences of same length.

268 209 Table 8.30 lists the summary of Tables 8.28 to Table 8.30 gives maximum number of sequences of given length s having magnitude of pairwise CCR value less than or equal to α. Table 8.30 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 16 ) by Expressing Element in GF(2 16 ) as 16 Tuple Length of Sequence s Value of x 0 Chaotic Map Equation and Table Number Maximum Number of Sequences of length s 0.6 g 6 Logistic Map Equation (8.2), Table g 3, g 6 Logistic Map Equation (8.1), Table g 3 Logistic Map Equation (8.2), Table g 2 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.2), Table g 3 Logistic Map Equation (8.2), Table g 9 Logistic Map Equation (8.2), Table g Logistic Map Equation (8.2), Table g Logistic Map Equation (8.1), Table g 5 Logistic Map Equation (8.2), Table g 6 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.1), Table g 9 Logistic Map Equation (8.2), Table g 9 Logistic Map Equation (8.1), Table g 4, g 6 Logistic Map Equation (8.2), Table g Logistic Map Equation (8.2), Table g 7 Logistic Map Equation (8.2), Table From the trials carried out and values listed in Tables 8.28 and 8.29 with r 1 = g and r 2 = g, it is seen that, for segments of length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 35 with x 0 = g 6 using Logistic Map Equation (8.2) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 24 with x 0 = g 3 or g 6 using Logistic Map Equation (8.1) as given in Table 8.27 and also with x 0 = g 3 using Logistic Map Equation (8.2) as given in Table 8.28.

269 210 iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 9 with x 0 = g 2 using Logistic Map Equation (8.1) as given in Table Similarly the maximum number of sequences are also listed for same length s with s = 31, 63, 127 and 255 in Table In the next section, CCR properties of chaotic binary sequences derived from sequence over GF(2 16 ) defined by Logistic map Equations (7.10) and (7.11) and by selecting a particular bit from each element in GF(2 16 ) is considered CCR Properties of Binary Sequences Generated by Selecting a Particular Bit From Each Element of the Sequence Over GF(2 16 ) Discrete chaotic sequence over GF(2 16 ) having maximum period using Logistic map Equation (8.1) and having maximum period using Logistic map Equation (8.2) are selected by computer simulation as discussed in Chapter 6. The elements in sequence are represented by binary 16 tuples. Binary sequence is obtained by selecting a particular bit from each element and concatenating the corresponding bits. For initial value x 0 = g, r 1 = g and r 2 = g, it is seen that the chaotic sequence over GF(2 16 ) defined by Logistic map given by Equation (8.1) has period Hence choosing bits in location i (i = 1 to 16) results in 16 sequences of length The CCR results of the binary sequences generated by selecting a particular bit from each element in GF(2 16 ) based on Logistic map given by Equation (8.1) for different α are summarized in Table 8.31.

270 211 Table 8.31 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Different Magnitude of Pairwise CCR Value α with x 0 = g, r 1 = g and r 2 = g Bit chosen i Number of segments having magnitude of pairwise CCR value α For segments of length 15 For segments of length 31 For segments of length 63 For segments of length 127 For segments of length MSB LSB For initial value x 0 = g, r 1 = g and r 2 = g, it is seen that the chaotic sequence over GF(2 16 ) defined by Logistic map Equation (8.2) has period The CCR results of the binary sequences generated by selecting a particular bit from each element in GF(2 16 ) using Equation (8.2) for different α are summarized in Table 8.32.

271 212 Table 8.32 Number of Segments of Different Lengths Derived from Sequence over GF(2 16 ) Defined by Logistic Map Equation (8.2) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Different Magnitude of Pairwise CCR Value α with x 0 = g, r 1 = g and r 2 = g Bit chosen i For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length MSB LSB From Table 8.31 and 8.32 it is seen that, the number of segments of sequences are more than the number of Gold sequences of same length. Also there are some segments of binary sequences having magnitude of pairwise CCR value less than that of Gold and large set Kasami sequences of same length. Table 8.33 lists the summary of Tables 8.31 to Table 8.33 gives maximum number of sequences of given length s having magnitude of pairwise CCR value less than or equal to α.

272 213 Table 8.33 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 16 ) by Selecting a Particular Bit from Element Over GF(2 16 ) Length of Sequence s Value of x 0 Chaotic Map Equation and Table Number Maximum Number of Sequences of length s 0.6 g 6 Logistic Map Equation (8.2), Table g 3 Logistic Map Equation (8.1), Table g 3 Logistic Map Equation (8.2), Table g 2 Logistic Map Equation (8.2), Table g Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.2), Table g 3 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.2), Table g 9 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.2), Table g Logistic Map Equation (8.2), Table g Logistic Map Equation (8.1), Table g 5 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.1), Table g 9 Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.2), Table g Logistic Map Equation (8.1), Table g 7 Logistic Map Equation (8.2), Table From the trials carried out and values listed in Tables 8.31 and 8.32 with r 1 = g and r 2 = g, it is seen that, for segments of length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 40 with x 0 = g 6 using Logistic Map Equation (8.2) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 25 with x 0 = g 3 using Logistic Map Equation (8.1) as given in Table 8.31 or using Logistic Map Equation (8.2) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 13 with x 0 = g 2 using Logistic Map Equation (8.2) as given in Table Similarly the maximum number of sequences are also listed for same length s with s = 31, 63, 127 and 255 in Table In the next section, CCR properties of chaotic binary sequences generated by mapping every field element in GF(2 16 ) to GF(2) is considered.

273 CCR Properties of Binary Sequences Generated by Mapping Every Field Element in GF(2 16 ) to GF(2) using Trace Function As discussed in Section using Trace function, every 16 tuple in GF(2 16 ) is mapped to distinct element in GF(2). The period of the binary sequence is in case of Logistic map Equation (8.1) and in case of Logistic map Equation (8.2). This binary sequence is divided in to non-overlapping segments of same length s, for s = 15, 31, 63, 127 and 255 to study the CCR properties. The number of non-overlapping segments of different lengths considered in this work is listed in Table 1.1. The pairwise CCR values of segment 1 with all other segments are computed using Equation (4.9). Segments having magnitude of pairwise CCR value less than a particular are selected. The CCR results of the binary sequences derived from sequence over GF(2 16 ) defined by Logistic map given by Equations (7.10) and (7.11) and by mapping every field element in GF(2 16 ) to GF(2) using trace function for different α are summarized in Table 8.34 and Table 8.35 respectively. Table 8.34 Number of Segments of Different Lengths Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.1) and using Trace Function having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g Trial No. x 0 chose n For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length g g g g g g g g g

274 215 The CCR results of the binary sequences generated by mapping every field element in GF(2 16 ) to GF(2) using trace function governed by Equation (8.2) for different α are summarized in Table Table 8.35 Number of Segments of Different Lengths Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.2) and using Trace Function having Different Magnitude of Pairwise CCR Value α with r 1 = g and r 2 = g Trial No. x 0 chose n For segments of length Number of segments having magnitude of pairwise CCR value α For segments For segments of For segments of of length 31 length 63 length For segments of length g g g g g g g g g From Table 8.34 and 8.35 it is seen that, the number of segments of sequences are more than the number of Gold sequences of same length. Also there are some segments of binary sequences having magnitude of pairwise CCR value less than that of Gold and large set Kasami sequences of same length. Table 8.36 lists the summary of Tables 8.34 to Table 8.36 gives maximum number of sequences of given length s having magnitude of pairwise CCR value less than or equal to α.

275 216 Table 8.36 Choice of initial value and chaotic map which gives maximum number of segments of given length and given peak magnitude of CCR value in case of binary sequences derived from sequence over GF(2 16 ) using Trace Function Length of Sequence s Value of x 0 Chaotic Map Equation and Table Number Maximum Number of Sequences of length s 0.6 g 5 Logistic Map Equation (8.1), Table g 6 Logistic Map Equation (8.2), Table g 2 Logistic Map Equation (8.1), Table g 7 Logistic Map Equation (8.2), Table g 6 Logistic Map Equation (8.1), Table g 6 Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.2), Table g 5 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.1), Table g 8 Logistic Map Equation (8.2), Table g 5 Logistic Map Equation (8.1), Table g 7 Logistic Map Equation (8.2), Table g 4 Logistic Map Equation (8.2), Table g 5 Logistic Map Equation (8.1), Table g 6 Logistic Map Equation (8.1), Table g Logistic Map Equation (8.1), Table g 2 Logistic Map Equation (8.2), Table g Logistic Map Equation (8.2), Table g Logistic Map Equation (8.1), Table g 6 Logistic Map Equation (8.2), Table 8.35 From the trials carried out and values listed in Tables 8.34 and 8.35 with r 1 = g and r 2 = g, it is seen that, for segments of length 15, i) Maximum number of segments having same CCR as Gold is equal to and is 32 with x 0 = g 5 using Logistic Map Equation (8.1) as given in Table 8.34 and also with x 0 = g 6 using Logistic Map Equation (8.2) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 24 with x 0 = g 2 using Logistic Map Equation (8.1) as given in Table 8.34 and also with x 0 = g 7 using Logistic Map Equation (8.2) as given in Table iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 12 with x 0 = g 6 using Logistic Map Equation (8.1) as given in Table 8.34 or using Logistic Map Equation (8.2) as given in Table 8.35.

276 217 Similarly the maximum number of sequences are also listed for same length s with s = 31, 63, 127 and 255 in Table Having discussed the properties of CCR, in the next section, LC properties of segments of sequences of length 15 exhibiting desirable pairwise CCR values are studied. 8.6 LC PROPERTIES OF SEGMENTS OF BINARY SEQUENCES OF LENGTH 15 DERIVED FROM CHAOTIC SEQUENCE GF(2 16 ) Based on the magnitude of pairwise CCR value α, segments of sequences are selected for three values of α. 0.6 corresponding to peak magnitude of CCR value of Gold sequences and large set Kasami sequences and two smaller values for which there are no Gold sequences or large set Kasami sequences. LC properties of segments of sequences of length 15 exhibiting desirable pairwise CCR values are computed. LC properties of non-overlapping segments of length 15 derived from sequence over GF(2 16 ) defined by Logistic map Equations (7.10) and (7.11) and by expressing filed elements in GF(2 16 ) as binary 16 tuples is considered next LC Properties of Binary Sequences of length 15 Generated by Expressing Field Element in GF(2 16 ) as Binary 16 Tuples To study the LC properties of segments of sequences derived from sequence over GF(2 16 ) defined by Logistic map Equation (8.1) and by expressing field elements in GF(2 16 ) as binary 16 tuples, set of sequences of length 15 listed in Table 8.28 is considered.

277 218 Segments of sequences of length s = 15 having magnitude of pairwise CCR value less than or equal to 0.6 as that of Gold sequences and two more lower values equal to 0.35 and 0.5 as tabulated in Table 8.28 is considered and their LC is computed. Out of 100 segments of length 15, a) 7 segments have magnitude of pairwise CCR value The LC of these 7 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 20 segments have magnitude of pairwise CCR value 0.5. The LC of these 20 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 26 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 26 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 9. Some of the segments of 15 have linear complexity 9, which is more than that of Gold sequence of same length. Table 8.37 Linear Complexity of 7 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

278 219 Table 8.38 Linear Complexity of 20 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.39 Linear Complexity of 26 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.1) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

279 220 Table 8.39 (Continued) LC properties of the segments of binary sequences derived from sequence over GF(2 16 ) defined by Logistic map Equation (8.2) and by expressing field elements in GF(2 16 ) as binary 16 tuples, set of sequences of length 15 which are listed in Table 8.29 is considered. Out of 100 segments of length 15, a) 8 segments have magnitude of pairwise CCR value The LC of these 8 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 22 segments have magnitude of pairwise CCR value 0.5. The LC of these 22 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 29 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 29 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10.

280 221 Table 8.40 Linear Complexity of 8 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.41 Linear Complexity of 22 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

281 222 Table 8.42 Linear Complexity of 29 Segments Derived from sequence over GF(2 16 ) Defined by Logistic Map Equation (8.2) and by Expressing Element in GF(2 16 ) as 16 tuple having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity In the next section, linear complexity properties of non-overlapping segments of length 15 generated using Logistic map given by Equations (8.1) and (8.2) by selecting a particular bit from each element of the sequence over GF(2 16 ) is considered.

282 LC Properties of Binary Sequences of Length 15 Generated by Selecting a Particular Bit from Each Element of the Sequence Over GF(2 16 ) Linear complexity values are computed for non-overlapping segments of length 15 exhibiting desirable pairwise CCR values which are derived from sequence over GF(2 16 ) using Logistic map Equation (8.1) and by selecting the MSB bit from each element of the sequence over GF(2 16 ) as listed in Table Out of 100 segments of length 15, a) 9 segments have magnitude of pairwise CCR value The LC of these 9 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 21 segments have magnitude of pairwise CCR value 0.5. The LC of these 21 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 34 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 34 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 9. Table 8.43 Linear Complexity of 9 Segments Derived from sequence over GF(2 16 ) using Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

283 224 Table 8.44 Linear Complexity of 21 Segments Derived from sequence over GF(2 16 ) using Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.45 Linear Complexity of 34 Segments Derived from sequence over GF(2 16 ) using Logistic Map Equation (8.1) and by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

284 225 Table 8.45 (Continued) To study the linear complexity properties of non-overlapping segments of length 15 generated using Equation (8.2) by selecting the MSB bit from each element of the sequence over GF(2 16 ), Table 8.32 is considered. Out of 100 segments of length 15, a) 9 segments have magnitude of pairwise CCR value The LC of these 9 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 25 segments have magnitude of pairwise CCR value 0.5. The LC of these 25 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 32 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 32 sequences are computed using

285 226 Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. Table 8.46 Linear Complexity of 9 Segments Derived using Logistic map Equation (8.2) by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.47 Linear Complexity of 25 Segments Derived using Logistic map Equation (8.2) by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

286 227 Table 8.47 (Continued) Table 8.48 Linear Complexity of 32 Segments Derived using Logistic map Equation (8.2) by Selecting a Particular Bit from Element Over GF(2 16 ) having Magnitude of Pairwise CCR Value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

287 228 LC properties of non-overlapping segments of length 15 defined by Equations (7.10) and (7.11) generated by mapping every field element in GF(2 16 ) to GF(2) using trace function is considered in the next section LC Properties of Binary Sequences Generated by Mapping Every Field Element in GF(2 16 ) to GF(2) Using Trace Function To study the LC properties, non-overlapping segments of length 15 exhibiting desirable pairwise CCR values which are derived from sequence over GF(2 16 ) defined by Logistic map Equation (8.1) and by mapping every field element in GF(2 16 ) to GF(2) using trace function, Table 8.34 is considered. Out of 100 segments of length 15, a) 10 segments have magnitude of pairwise CCR value The LC of these 10 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 20 segments have magnitude of pairwise CCR value 0.5. The LC of these 20 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 30 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 30 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10.

288 229 Table 8.49 Linear Complexity of 10 Binary Sequences Derived using Logistic Map Equation (8.1) and Trace Function having Magnitude of Pairwise CCR value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.50 Linear Complexity of 20 Binary Sequences Derived using Logistic Map Equation (8.1) and Trace Function having Magnitude of Pairwise CCR value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

289 230 Table 8.51 Linear Complexity of 30 Binary Sequences Derived using Logistic Map Equation (8.1) and Trace Function having Magnitude of Pairwise CCR value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity To study the linear complexity properties of non-overlapping segments of length 15 generated using Equation (8.2) by mapping every element in GF(2 16 ) to GF(2) using trace function, set of sequences having desirable pairwise CCR values which are listed in Table 8.35 is considered.

290 231 Out of 100 segments of length 15, a) 9 segments have magnitude of pairwise CCR value The LC of these 9 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table b) 22 segments have magnitude of pairwise CCR value 0.5. The LC of these 22 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table c) 29 segments have magnitude of pairwise CCR value 0.6 corresponding to that of Gold and large set Kasami sequences. The LC of these 29 sequences are computed using Berlekamp Massey algorithm. The segment numbers, sequences and LCs are listed in Table In all the above cases it is seen that LC value lies in the range 8 and 10. Table 8.52 Linear Complexity of 9 Binary Sequences Derived using Logistic Map Equation (8.2) and Trace Function having Magnitude of Pairwise CCR value 0.35 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

291 232 Table 8.53 Linear Complexity of 22 Binary Sequences Derived using Logistic Map Equation (8.2) and Trace Function having Magnitude of Pairwise CCR value 0.5 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity Table 8.54 Linear Complexity of 29 Binary Sequences Derived using Logistic Map Equation (8.2) and Trace Function having Magnitude of Pairwise CCR value 0.6 along with Segment Number Serial Number Segment Number Binary Sequence Linear complexity

292 233 Table 8.54 (continued) Having discussed the CCR and LC properties of chaotic binary sequences derived from Chaotic sequences over GF(2 16 ), in the next section, the results are discussed in brief. 8.7 RESULTS AND DISCUSSION It is seen that the normalized cyclic HACR function of discrete sequence over GF(2 8 ) or or GF(2 16 ) has two values, + 1 for shift τ = 0 and -1 for τ 0. The balance property is satisfied for some choice of initial values x 0 and bifurcation parameters r 1 and r 2 in case of chaotic sequences over GF(2 8 ) or GF(2 16 ), generated using Logistic map Equation (8.2). In these cases all the elements in the sequence occur only once.

293 234 Table 8.55 Choice of Initial value and Chaotic Map which gives Maximum Number of Segments of given length and given Peak Magnitude of CCR value in case of Binary Sequences Derived from Sequence over Finite Field Length of Sequence Value of x 0 Method of Deriving Binary Sequence Chaotic Map Equation and Table Number Maximum Number of Sequences g 6 Logistic Map Equation (8.2), Table g 3 Logistic Map Equation 0.5 By selecting a particular bit (8.1), Table g 3 from element of the sequence over GF(2 16 ) Logistic Map Equation (8.2), Table g 2 Logistic Map Equation (8.2), Table g By expressing field element Logistic Map Equation 55 in GF(2 16 ) as binary 16 tuples (8.2), Table g 4 from element of the sequence (8.2), Table By selecting a particular bit Logistic Map Equation over GF(2 16 ) By expressing field element Logistic Map Equation g 3 g 3 g 9 g 9 g g g in GF(2 16 ) as binary 16 tuples By selecting a particular bit from element of the sequence over GF(2 16 ) By expressing field element in GF(2 16 ) as binary 16 tuples By selecting a particular bit from element of the sequence over GF(2 16 ) By expressing field element in GF(2 16 ) as binary 16 tuples By selecting a particular bit from element of the sequence over GF(2 16 ) g 4 element in GF(2 16 ) to GF(2) By mapping every field using Trace function g 5 from element of the sequence By selecting a particular bit over GF(2 16 ) g By expressing field element in GF(2 16 ) as binary 16 tuples g 9 from element of the sequence By selecting a particular bit over GF(2 16 ) g 2 element in GF(2 16 ) to GF(2) By mapping every field using Trace function g By selecting a particular bit from element of the sequence over GF(2 16 ) g 7 from element of the sequence By selecting a particular bit over GF(2 16 ) (8.2), Table 8.28 Logistic Map Equation (8.2), Table 8.32 Logistic Map Equation (8.2), Table 8.28 Logistic Map Equation (8.1), Table 8.31 Logistic Map Equation (8.2), Table 8.32 Logistic Map Equation (8.2), Table 8.28 Logistic Map Equation (8.2), Table Logistic Map Equation (8.2), Table Logistic Map Equation (8.1), Table Logistic Map Equation (8.1), Table Logistic Map Equation (8.2), Table Logistic Map Equation (8.2), Table Logistic Map Equation (8.1), Table Logistic Map Equation (8.2), Table

294 235 Table 8.55 summarizes Tables 8.3, 8.6, 8.9, 8.30, 8.33 and It gives the choice of Logistic maps with different initial values from GF(2 m ) and binary maps which gives maximum number of segments of binary sequences derived from sequence over finite field and of different length s, having peak magnitude of pairwise CCR value less than or equal to three chosen values of α. First column in Table 8.56 gives the length of the sequence, second column gives magnitude of pairwise CCR values chosen, third column gives values of n, fourth column gives method of deriving binary sequence, fifth column lists the type of chaotic map and corresponding Table number from which maximum number is obtained. The last column gives the maximum number of sequences. From Table 8.55, it is seen that for segment of length 15, i) Maximum number of segments having same CCR as Gold sequence is equal to 40 with x 0 = g 6 using Logistic Map Equation (8.2) by selecting a particular bit from element of the sequence over GF(2 16 ) as given in Table ii) Maximum number of segments having magnitude of pairwise CCR value 0.5 is equal to 25 with x 0 = g 6 using Logistic Map Equation (8.1) or Logistic Map Equation (8.1) and by selecting a particular bit from element of the sequence over GF(2 16 ) as given in Table 8.31 and Table 8.32 respectively. iii) Maximum number of segments having magnitude of pairwise CCR value 0.35 is equal to 13 with x 0 = g 2 using Logistic Map Equation (8.2) by selecting a particular bit from element of the sequence over GF(2 16 ) as given in Table For length 31, the maximum number of segments having same CCR as Gold sequence is equal to 55 with x 0 = g using Logistic Map Equation (8.2) by expressing field element in GF(2 16 ) as binary 16 tuples as given in Table For length 127, the maximum number of segments having same CCR as Gold sequence is equal to and is 236 with x 0 = g 5 using Logistic Map Equation (8.1) and by selecting a particular bit from element of the sequence over GF(2 16 ) as given in Table 8.31.

295 236 For length 255, the maximum number of segments having same CCR as Gold sequence is equal to and is 433 with x 0 = g 6 using Logistic Map Equation (8.2) by mapping every field element in GF(2 16 ) to GF(2) using Trace function as given in Table Similarly it is shown for other cases in Table It is seen that from Table 8.55, it is possible to obtain segments of sequences of same length s, for s = 15, 31, 63, 127 and 255 having magnitude of pairwise CCR value same as that of Gold or large set Kasami sequences of same length and the number of sequences is almost double the number of Gold sequences of same length. It is also seen that the LC of some segments of binary sequences of length 15 is equal to 11 with x 0 = g using Logistic Map Equation (8.2) by expressing field element in GF(2 8 ) as binary 8 tuples as given in Table It is observed that binary sequences derived from sequences over GF(2 16 ) have larger period and give more number of segments exhibiting desirable CCR and LC properties compared to sequences derived from sequences over GF(2 8 ). The correlation bounds and BER performance is studied in the next chapter. 8.8 SUMMARY The chaotic discrete sequences over GF(2 m ) using Logistic map equations and deriving chaotic binary sequence from them using three methods are proposed. Here two standard word sizes of m = 8 and m = 16 are considered. It is seen that elements over GF(2 m ) have low values of HACR and good balance properties. CCR and LC properties are investigated for 15 bit binary sequences derived from elements over GF(2 8 ) using two Logistic map equations and compared with Gold sequences and large set Kasami

296 237 sequences of same length. From the results it is seen that there are some segments of 15 bit having magnitude of pairwise CCR value less than that of Gold sequences and linear complexity varying between 8 and 11. Some of the segments of 15 have linear complexity more than that of Gold sequences. The investigation is extended to binary sequences derived from sequence over GF(2 16 ). CCR values of segments of sequences of same length s, for s = 15, 31, 63, 127 and 255 also found to have desirable properties. Number of segments of binary sequences having magnitude of pairwise CCR values same as that of Gold sequences are more than the number of Gold sequences of same length, in this case compared to the case of GF(2 8 ). More number of segments of sequences compared to the case of GF(2 8 ), exhibit desirable CCR and LC properties. There are some segments of binary sequences of same length s, for s = 15, 31, 63, 127 and 255 with good cross correlation properties as compared to Gold sequences and large set Kasami sequences of same length. The linear complexity of some of the sequences is found to be larger than that of Gold and large set Kasami sequences.

297 CHAPTER 9 CORRELATION BOUND AND PERFORMANCE EVALUATION

298 238 CHAPTER 9 CORRELATION BOUND AND PERFORMANCE EVALUATION There are several bounds on the cross correlation of sequences known. The most commonly used are Welch and Sidelnikov bound. The impact of these bounds is that they dictate the limits within which all code sequence designs must lie. Thus it is not possible to independently design the correlation value and the set size, but it is necessary to allow the increase of the maximum absolute value of the correlation value in order to increase the set size for given sequence length (Kimmo Kettunen 1997). As an application of proposed segments of binary sequences having low pairwise CCR values in communication applications, DS-CDMA system is considered. The correlation bound of proposed segement of binary sequences is considered in the next section. 9.1 CORRELATION BOUND Set of periodic sequences with good correlation properties are required for variety of applications and much effort has been spent on design of such sequences. There is a fundamental limit on how the correlation can be for a family of sequences. The most efficient bounds are those due to Welch and Sidelnikov (Helleseth T. and Kumar P. V.

299 ) and have been used as bench mark for testing sequences. A lower bound to the maximum cross correlation value for a family of complex spreading sequences was first derived by Welch (Welch L.R. 1974). Around the same time, Sidelnikov (Sidelnikov V.M. 1971) derived a lower bound on the maximum value of cross correlation of such sequences. Both Welch bound and Sidelnikov bound have been used extensively in the design and analysis of sequences for CDMA (Sarwate D.V. and Pursley M.B. (1980), Rowe H.E. (1982), Jong-Seon No and Kumar P.V. (1989), Kumar P.V. and Liu C.M. (1990), Kumar P.V. and Moreno O. (1991) and Udaya P. and Siddiqi M.U. (1996)). The Welch bound and Sidelnikov bound provide lower bounds on non-trivial normalized peak magnitude cross correlation value denoted by. Welch s bound has proved extremely useful in many applications, in particular for evaluating the quality of sequence sets for use in CDMA (Massey J.L. 1991). In CDMA systems, there is greatest interest in designs in which the parameter is in the range (Helleseth T. and Kumar P. V. 1999), where N is the period of the binary sequence. Welch bound is considerd next Welch Bound Welch developed (Welch L.R. 1974) the lower bound of the cross correlation (normalized with respect to N) between any pair of binary sequences of period N in the set of γ sequences and is given by, (9.1) For large values of γ i.e., about ten or more, (Valery P. Ipatov 2005), Equation (A 3.1) becomes, (9.2) Sidelonikov bound is defined next.

300 Sidelnikov Bound The Sidelnikov bound (normalized with respect to N) states that for any γ N, (9.3) The maximum cross correlation value for small set of Kasami sequences coincides with this Welch lower bound and thus it is optimal. For large set Kasami sequences, the lower bound of the cross correlation is better described by the Sidelnikov bound (Byeong Gi Lee and Byoung Hoon Kim 2002). Gold sequences do not achieve Welch lower bound but the cross correlation values are better with respect to Sidelnikov bound for odd number of stages (Sarwate D.V. and Pursley M.B. 1980). Udaya - Siddiqi (US) sequences satisfy both Welch bound and Sidelnikov bound with equality (Udaya P. and Siddiqi M.U. 1996). Table 9.1 gives the correlation bound of Gold and Kasami sequences for the binary sequences of length 15, 31, 63, 127 and 255. It is shown that Gold sequences and large set Kasami sequences are suboptimal with respect to Sidelnikov bound and small set Kasami sequences are optimal with respect to Welch bound. Consider Table 7.9 which lists the number of segments of binary sequences of length 15, 31, 63, 127 and 255 which are derived from Logistic map given by Equation (7.1) and mapping P 0 for different values of n and for three different values of. For each value of and segments of sequence of given length, set of sequences having maximum number of segments is considered to apply the correlation bound. The values are computed and tabulated in Table 9.2.

301 241 Table 9.1 Correlation Bound for segments of sequences of Gold and Kasami Sequences Sl. No N γ Gold Sequences α max Welch bound Small Set Sidelnik Welch ov bound γ α max bound Kasami Sequences Sidelnik ov bound γ α max Large Set Welch bound Sidelnik ov bound Table 9.2 Correlation Bound for segments of sequences derived from Logistic map Equation (7.1) and mapping P 0 Sl. No. N γ For Welch bound For Sidelnikov bound (with γ N) From Table 9.2, it is found that a smaller set of Segments of binary sequences of length 15, 31, 63, 127 and 255 having much lower than CCR values of Gold and large set Kasami sequences, satisfies the Welch bound and Sidelnikov bound better than Gold and large set Kasami sequences of same length. Similar results are seen for the

302 242 segments of sequences in Table 7.10 for Logistic map Equation (7.2) and mapping P 0 and are tabulated in Table 9.3. Table 9.3 Correlation Bound for segments of sequences derived from Logistic map Equation (7.2) and mapping P 0 Sl. No. N γ For Welch bound For Sidelnikov bound (with γ N) It is shown that a smaller set of proposed segments of binary sequences of length 15, 31, 63, 127 and 255 derived from chaotic sequences over Z 4 using mapping P 0 having much lower than CCR values of Gold and large set Kasami sequences, satisfies the Welch bound and Sidelnikov bound better than Gold and large set Kasami sequences of same length. It is verified for smaller set of proposed segments of binary sequences of same lengths 15, 31, 63, 127 and 255 derived from chaotic sequences over Z 4 using other mapping P 1, P 2, P 3 and binary sequences derived from chaotic sequence over GF(2 m )

303 243 having much lower than CCR values of Gold and larges set Kasami sequences, also satisfies the Welch bound and Sidelnikov bound better than Gold and large set Kasami sequences of same length. The performance evaluation of the proposed segments of binary sequences is considered in the next section. 9.2 PERFORMANCE EVALUATION BASED ON PEAK MAGNITUDE OF PAIRWISE CCR VALUE To investigate the suitability of binary sequences with low pairwise CCR values, generated by proposed schemes in communication applications, DS-CDMA system is considered (Theodore S. Rappaport 2002). The users binary data sequences are denoted by m i (t), i = 1, 2, The spreading sequences are segments of binary sequences derived from sequence over Z 4 or GF(2 8 ) or GF(2 16 ) and are denoted by a i (t), i = 1, 2, The binary sequence with elements 0 and 1 is converted to 1 or 1 in both sequences m i (t) and a i (t) and binary PSK modulation is assumed. To recover m j (t) at the j th receiver coherent detection of BPSK is adopted. The despreading sequence is a j (t). A simplified diagram of a K-user CDMA system is presented in Figure 9.1. m 1 (t) S 1 (t) a 1(t) cos(ω 0t+Φ 1) r (t) (.)dt Z i (j) Threshold m j * (t) m K (t) S K (t) cos(ω 0t+Φ j) a j(t) a K(t) cos(ω 0t+Φ K) Transmitting part for K users Receiving part for user j Figure 9.1 Simplified Diagram of a CDMA System.

304 244 The signal S i (t) is originating from users i, i = 1, 2, K and r(t) is the signal at the input of a receiver. It is shown in Vladeanu C. et al (2001), for a synchronous system that assumes synchronized spreading sequences and constant power levels for all K users, the average BER depends on the white noise Power Spectral Density (PSD) and Multiple Access Interference (MAI) power. The average BER, P e is given by, (9.4) Where E b = P j T is the signal energy per bit period, N 0 /2 is the double sided white noise power spectral density, E b /N 0 is the signal-to-noise ratio, K is number of users, is the mean square cross-correlation value and N is the length of the spreading sequence. The derivation of Equation (9.4) is given in Appendix 1. The Q function is defined by integral, (9.5) Q(x) is a monotonically decreasing function with increase in its argument x. Values of Q(x) for different x are available as table (Simon Haykin 1989). The CCR values between two sequences of same length depend on the two sequences considered and the shift. Smaller the CCR values better the immunity against MAI. To study the performance, the maximum magnitude of pairwise CCR value α is considered and used in Equation (9.4). It is seen from Equation (9.4) and property of Q( ) function that the average BER of a synchronous DS-CDMA system versus the number of users depends on the MAI term. The MAI contribution to BER grows with the number of simultaneous users K in the system and is small if CCR value of spreading sequences is low. For a given number

305 245 of users K, using spreading sequences having lower maximum magnitude of pairwise CCR value results in lower BER. For a given BER the number of users can be increased if sequences with low CCR values are used. Since α is considered to be normalized magnitude of pairwise CCR value, substituting = α max 2 in Equation (9.4), gives bit error rate under worst conditions. Therefore Equation (9.4) can be written as, (9.6) Equality in Equation (9.6) is considered here for comparison under worst case. 2 Taking into consideration α max value, the BER performance given by Equation (9.6) for the binary sequences generated using six chaotic map equations defined over reals and two chaotic map equations defined over GF(2 8 ) and GF(2 16 ) are investigated. In all these cases it is observed that the number of sequences having peak magnitude of pairwise CCR value equal to that of Gold sequences is almost double the number of Gold sequences of same length. There are also set of sequences whose peak magnitude of pairwise CCR value is less than that of Gold sequences or large set Kasami sequences of same length. The number of such sequences depends on the chaotic map equations and binary mapping techniques. The number K of simultaneous users is less than or equal to the number of sequences. BER performances given by Equation (9.6) for the proposed sequences are compared with Gold sequences and large set Kasami sequences. It is found that as number of user s increases, BER also increases for a fixed value of. However using the proposed binary sequences as code sequences whose maximum magnitude of pairwise CCR value is less than that of Gold sequences, the BER performance is improved.

306 246 Figure 9.2 Variation of BER with Number of Users for Sequences of Length 15 and = 8 db. Figure 9.2 shows the increase of BER with number of users, using segments of sequences of length 15 derived from sequence over Z 4 or GF(2 8 ) or GF(2 16 ). From Figure 9.2, it is seen that, using set of sequences whose maximum magnitude of pairwise CCR value less than that of Gold sequences and large set Kasami sequences, the BER decreases and hence performance is improved. For = 8 db and for a given BER = 0.04, using Gold sequences and large set Kasami sequences, the number of users is only 3. Using proposed sequences with magnitude of pairwise CCR value 0. 5, the number of users are increased to 4 and with magnitude of pairwise CCR value 0.35 using the proposed sequences, the number of users are increased to 6. It is possible to obtain segments of chaotic binary sequences whose pairwise CCR value is low. Using such sequences as code sequences for a given BER and, the number of users can be increased. For given number of users = 9 and = 8 db, the BER for Gold and large set Kasami sequences is 0.2, using proposed sequences with pairwise 0. 5 the BER is 0.15 and the BER is 0.08 using proposed sequences with pairwise 0.35.

307 247 Figure 9.3 depict the variation of BER with with for the sequences of length 15. From Figure 9.3, it is seen that, for 5 simultaneous users with = 8, the BER is found to be 0.15 with 0.6, 0.08 for 0.5 and 0.03 for 0.35 respectively. Figures 9.4, 9.6, 9.8 and 9.10 show the plots of variation of BER with number of users for sequence generated using proposed sequences for segments of length 31, 63, 127 and 255 respectively. Here also it is seen that using set of sequences whose maximum magnitude of pairwise CCR value less than that of Gold sequences and large set Kasami sequences BER performance improves. Figures 9.5, 9.7, 9.9 and 9.11 show the plots of variation of BER with for the sequences of length 31, 63, 127 and 255 respectively. From these figures, it is evident that it is possible to get improvement in BER performance. Figure 9.3 Variation of BER with for Sequences of Length 15 and Number of Simultaneous Users = 5.

308 248 Figure 9.4 Variation of BER with Number of Users for Sequences of Length 31 = 8dB. Figure 9.5 Variation of BER with for Sequences of Length 31 and Number of Simultaneous Users = 9.

309 249 Figure 9.6 Variation of BER with Number of Users for Sequences of Length 63 = 8dB. Figure 9.7 Variation of BER with for Sequences of Length 63 and Number of Simultaneous Users = 20.

310 250 Figure 9.8 Variation of BER with Number of Users for Sequences of Length 127 = 8dB. Figure 9.9 Variation of BER with for Sequences of Length 127 and Number of Simultaneous Users = 40.

311 251 Figure 9.10 Variation of BER with Number of Users for Sequences of Length 255 = 8dB. Figure 9.11 Variation of BER with for Sequences of Length 255 and Number of Simultaneous Users = 60.

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