Excellent Performances of The Third-level Disturbed Chaos in The Cryptography Algorithm and The Spread Spectrum Communication

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1 Joural of Iformatio Hidig ad Multimedia Sigal Processig c 26 ISSN Ubiquitous Iteratioal Volume 7, Number 4, July 26 Excellet Performaces of The Third-level Disturbed Chaos i The Cryptography Algorithm ad The Spread Spectrum Commuicatio Jua Wag,2 ad Qu Dig Electroic egieerig istitute, Heilogjiag Uiversity, Harbi, Chia 2 Electroic ad iformatio egieerig istitute, Heilogjiag Uiversity of sciece ad techology, Harbi, Chia @qq.com, qudig@aliyu.com Received April, 26; revised May, 26 Abstract. I this paper, a ew logistic chaos was obtaied by improvig the traditioal logistic chaos, ad the secod-level ad the third-level chaoses were costructed i view of the ideas of cascade ad disturbace. The performaces of dyamic characteristics ad statistical radomess were simulated ad aalyzed for the chaotic sequeces at all levels, ad the third-level disturbed chaotic sequece was proved to be more excellet. Cosequetly, whe the third-level disturbed chaotic sequece is itroduced ito the cryptography algorithm ad the spread spectrum commuicatio, improved performace may be achieved. Keywords: Chaos; Cascade; Disturbace; Dyamic characteristic; Statistical radomess.. Itroductio. Chaos is a determiistic ad pseudo-radom process produced by the oliear dyamic system. Chaos caot be coverget but has bouds, it is aperiodic ad sesitively depedet to the iitial value []. Chaotic attractor has the characteristics of topological trasitivity ad miscibility ad it is sesitive to the, as a result it fits for the cofusio priciple i cryptography desig. Meawhile, the divergece of trajectories ad the sesitivity to iitial value of chaos fit for the diffusio priciple i cryptography desig. Therefore, chaos has gradually become the focus of cryptography research [2]. O the other had, chaos ca provide umerous pseudo-radom reewable sigals, which are o-correlative ad determiistic. Whe chaos is applied to the spread spectrum commuicatio, the performaces i ati-oise, ati-iterferece, ati-fadig, ati-multipath ad ati-decipher would be improved too [3, 4]. With the developmet of chaotic cryptography ad chaotic commuicatio techology, the requiremets i the applicatios of chaotic sequeces have bee icreasig. There are a series of disadvatages for the traditioal logistic chaos, such as small surjective iterval, arrow parameter rage ad low complexity, etc. I this paper, the traditioal logistic chaos was improved ad a ew logistic chaos was obtaied. I additio, the secod-level ad third-level chaoses were costructed i view of the ideas of cascade ad disturbace. The performaces of dyamic characteristics ad statistical radomess were tested ad aalyzed for the chaotic sequeces at all levels, ad reached the coclusio that the 826

2 Excellet Performaces of The Third-level Disturbed Chaos i The Cryptography Algorithm 827 dyamic ad pseudo-radom characteristics of the third-level disturbed chaotic sequece are more excellet. Whe the third-level disturbed chaotic sequece is used as a key or a spread spectrum sequece, the cryptography algorithm ad the spread spectrum commuicatio would obtai better performace. 2. The first-level chaos. the discrete chaotic map is fast to iterate ad easy to cotrol, it has icomparable superiority compared with the cotiuous chaotic system [5]. At preset, the applicatio research of chaos is mostly focused o the discrete chaotic map, which ca be described by differece equatio. 2.. The first-level traditioal chaos. The equatio of the first-level traditioal logistic map is give to be, x + = µx ( x ) =, 2, 3... () I Eq. (), x is the value of beig iterated times where x (, ], ad x + is the value of beig iterated + times; the iterval of is µ (, 4], while whe µ ( , 4], the traditioal should be i the chaotic state. As a kid of chaos, the traditioal logistic chaos has bee widely researched ad applied owig to its simple mathematical model. Although the traditioal logistic chaos has preferable performaces of radomess ad correlatio, ad its circuit is easier to be implemeted. the disadvatages of the traditioal logistic chaos are also apparet, such as ifiite fixed-poit attractor, small surjective iterval, arrow parameter rage ad low complexity, etc [6, 7, 8, 9]. Whe the is improperly chose, the iteratio results would ted to be a fixed-value or ted to be collectively distributed, the security of cryptography algorithm or the ati-iterferece of spread spectrum commuicatio would be iflueced to a certai extet. Therefore, the traditioal logistic chaos is ecessary to be improved The first-level ew chaos. The equatio of the first-level ew is give to be, x + = µx ( x 2 )mod =, 2, 3... (2) I Eq. (2), the iterval of is µ (, 4], the iterval of iitial value is x (, ], ad mod is the stadard modulo operatio. From Eq. (2) we ca see that µx ( x 2 ) is greater tha zero. I the operatio of mod, Whe µx ( x 2 ) is a iteger, the value of x + is zero; whe µx ( x 2 ) is ot a iteger, the value of x + is a decimal. the origial iteratio values beyod the scope of the defiitio domai ca be decreased, fallig ito the iterval (, ] ad iterated agai. Although there is oly a trivial improvemet, the disadvatages i the traditioal logistic chaos may be effectively ameliorated. 3. The secod-level chaos. At preset, there is ot a possible theoretical framework o how to sigificatly improve the dyamic characteristics ad pseudo-radomess of the discrete chaotic map []. Herei, based o the ideas of cascade ad disturbace, we proposed several solutios i this paper. 3.. The secod-level cascaded chaos. The model of is show i figure. The first iteratio output of ew is iput as the iitial value ito the traditioal, ad the first iteratio output of traditioal logistic map is iput ito the ew for the ext iteratio. Such cycled process ca produce the secod-level cascaded chaotic sequece.

3 828 J. Wag ad Q. Dig the ew the traditioal Figure. Model of Substitutig x i Eq. () to be x + i Eq. (2), The equatio of the secod-level cascaded chaos is give to be, x + = µx ( x ) =, 2, 3... (3) I Eq. (3), the iterval of is µ (, 4]; the iitial value is give to be x = µx ( x 2 ) mod, where x (, ] The secod-level disturbed chaos. The model of is show i figure 2, the of ew is disturbed by the iteratio output of traditioal, ad the iteratio output of ew is the secod-level disturbed chaotic sequece. the traditioal the ew disturbe Figure 2. Model of Substitutig µ i Eq. (2) to be 4x +, ad x + is i Eq. (). The equatio of the secod-level cascaded chaos is give to be, x + = µ x ( x 2 )mod =, 2, 3... (4) I Eq. (4), the iterval of iitial value is x (, ]; the is give to be µ = 4µx ( x ), where µ (, 4]. 4. The third-level chaos. I order to improve the complexity ad the dyamics characteristics, achieve larger parameter ad surjective iterval for the discrete chaotic map, i this paper, the third-level chaos was costructed o the basis of the secod-level chaos ad the improved tet chaos. 4.. The third-level cascaded chaos. The model of is show i figure 3, the ew, the traditioal ad the improved tet map were cascaded to costruct, the equatio of improved tet map is give to be [], x + = abs( µx ) =, 2, 3... (5) I Eq. (5), the iterval of iitial value is x (, ]; the iterval of is µ (, 2], Whe µ [, 2], the improved tet map should be i chaotic state.

4 Excellet Performaces of The Third-level Disturbed Chaos i The Cryptography Algorithm 829 Substitutig x i Eq. (5) to be x + i Eq. (3), The equatio of the third-level cascaded chaos is give to be, x + = abs [ µ µ 2 x ( x )] =, 2, 3... (6) I Eq. (6), µ is the of improved tet map, where µ (, 2]; µ 2 is the of, where µ 2 (, 4]; the iitial value is give to be x = µx ( x 2 )mod, where x (, ]. the ew the traditioal the improved tet map Figure 3. Model of 4.2. The third-level disturbed chaos. The model of is show i figure 4, the of ew is disturbed by the iteratio output of traditioal, ad the iteratio iput of ew is disturbed by the iteratio output of improved tet map. Correspodigly, the iteratio output of ew is the third-level disturbed chaotic sequece. Substitutig x i Eq. (4) to be x + i Eq. (5), The equatio of the third-level disturbed chaos is give to be, [ x + = µ x (x ) 2] mod =, 2, 3... (7) I Eq. (7), the is give to be µ = 4µx ( x ), where µ (, 4]; the iitial value is give to be x = abs( µx ), where x (, ]. the improved tet map disturbe iitial value the traditioal the ew disturbe Figure 4. Model of 5. Performace aalysis. 5.. Performace of the. may be adopted to quatitatively characterize the overall effect of approximatio or separatio betwee the trajectories geerated by the oliear map. For the oliear map, the Lyapuov expoet represets the average expoet divergece rate of trajectories i the directio of each basic vector i -dimesioal phase space. Whe is egative, the distace betwee the trajectories disappears expoetially, The motio state of the system correspods to a periodic motio or fixed-poit. Whe the is positive, the trajectories which are adjacet i the iitial state would separate expoetially, The motio of the system correspods to chaotic state. Whe is zero, the distace betwee the trajectories is costat, ad the iteratio poit correspods to the bifurcatio poit, i.e., the positio where the period is doubled [2].

5 83 J. Wag ad Q. Dig The s of chaos at all levels are show i figure 5. As ca be see from the simulatio, the of traditioal logistic chaos is positive oly whe µ > 3.57, ad the egative parameter poits of would emerge repeatedly i this rage. For the ew logistic ad multi-level chaoses, the parameter rage correspods to a positive is obviously elarged. I particular, the is larger ad stable for the secod-level ad third-level disturbed chaoses. It may be see i compariso that is more sophisticated, more difficult to predict, ad the trouble that the fixed ad periodic poits appear i the chaotic parameter rage may be overcome effectively Figure 5. s of chaos at all levels 5.2. Sesitivity of the iitial value. The radomess of chaotic sequeces lies i the sesitivity of iitial value. Whe two iitial values which are very close are iputted ito the chaotic map, after beig iterated may times, the subsequet trajectories would be separated completely. The property that a tiy differece i iitial value would lead to a large differece i output is referred to as iitial value sesitivity [3]. The iitial value sesitivities of chaos at all levels are show i figure 6, i which two iteratio sequeces x ad x2 are selected from each chaotic map, ad the iitial values of x ad x2 are set to.6 ad.6. Whe the iteratio times are of the same, a large differece would emerge after beig iterated may times, eve though there is oly a tiy differece betwee the iitial value of x ad x2. I particular, the secodlevel ad the third-level chaoses are more sesitive to the iitial value. As a result a large amout of chaotic sequeces would be geerated ad the sequece space would be effectively elarged, makig the cryptaalysis more complicated Performace of the bifurcatio. Bifurcatio diagram is ofte used to ituitively reflect the two-dimesioal relatios betwee the ad the umerical distributio of chaotic sequeces. Based o the bifurcatio diagram, the whole process of the chaotic map eterig the chaotic state may be ituitively observed [4].

6 Excellet Performaces of The Third-level Disturbed Chaos i The Cryptography Algorithm 83 The bifurcatio characteristics of chaos at all levels are show i figure 7. As ca be see from the simulatio, the iteratio poits of traditioal logistic chaos would be mapped to the whole iterval [, ] for a of µ = 4, kow as surjective map. With the icreasig of the, the bifurcatios of ew logistic ad multi-level chaoses become more ad more itesive, ad the parameter rage of surjective map is gradually exteded. I particular, may completely eter ito the chaotic state. As a result, the parameter selectivity of surjective map ad the dyamic characteristics of chaotic sequece are simultaeously improved, ad the problem of stable widow is effectively solved Performace of the attractor. A group of trajectories i phase space is ofte referred to as a attractor, by which the overall system stability ca be maifested. The structure ad complexity of a attractor determie the dyamic characteristics of chaos, so it is a importat way to study chaos [5]. Two-dimesioal attractors of chaos at all levels are show i figure 8. As ca be see from the simulatio, the traditioal logistic chaos has a helmet-shaped attractor, while the phase space distributios of ew logistic ad multi-level chaoses are more disorderly scattered, the differece betwee adjacet iteratio values is larger, ad the iteratio may be effectively avoided tedig to the same value. it is apparet that the attractor structure of possesses a higher level of complexity, it does ot have ay regularities, effectively reducig the possibility of sequece beig deciphered Performace of the ergodicity. Ergodicity idicates the radomess of chaotic sequece i the correspodig value rage. The radomess ad uiformity of chaotic sequece may be ituitively reflected by the distributio of chaotic sequece i the whole iteratio iterval [6]. Ergodicities of chaos at all levels are show i figure 9, the ad the iitial value were set to be µ = 3.56 ad x =., respectively. As ca be see from.9 x x2.9 x x2.9 x x x x2.9 X X2.9 x x Figure 6. Iitial value sesitivities of chaos at all levels

7 832 J. Wag ad Q. Dig umerical distributio umerical distributio umerical distributio umerical distributio umerical distributio umerical distributio Figure 7. The bifurcatio characteristics of chaos at all levels x x x x x x x x x x x x Figure 8. Two-dimesioal attractors of chaos at all levels the simulatio, the iteratio distributio of traditioal logistic chaos is ot uiform i the iterval[, ]. Whe the chaos is improved ad its level is icreased, the values of x would etirely traverse the iterval [, ] ad its distributio would ted to be uiform. I cotrast, the ergodicity of is more ideal, idicatig that it has better statistical radomess.

8 Excellet Performaces of The Third-level Disturbed Chaos i The Cryptography Algorithm Figure 9. The ergodicities of chaos at all levels 5.6. Performace of the histogram. Histogram idicates the radomess of chaotic sequece i the correspodig value rage. Histograms of chaos at all levels are show i Figure, the ad the iitial value were set to be µ = 3.98 ad x =., respectively. As ca be see from the simulatio, the histogram become uiform whe the chaos is improved ad its level is icreased. Comparatively speakig, the histogram of is more ideal. This suggests that the third-level disturbed chaos has excellet statistical radomess to resist the statistical aalysis attack Performace of the correlatio. The relatioship betwee the correlatio performace of chaotic sequece ad the security of cryptography algorithm or the atiiterferece of the spread spectrum commuicatio are compact. The expressio of selfcorrelatio fuctio is give to be = N x x +m (8) N Self-correlatio performaces of chaos at all levels are show i Figure, the system parameter ad the iitial value were set to be µ = 3.56 ad x =., respectively. Whe the chaos is improved ad its level is icreased, the iteratio values are more radomly distributed i a larger scope ad the sequece space would be elarged. Amog them, the self-correlatio fuctio of is close to the ideal impulse fuctio, which idicates the statistical radomess of is excellet. 6. Coclusio. Based o the compariso ad aalysis of the dyamic characteristics idicated by the, attractor, etc, we foud that the third-level disturbed chaos has a larger surjective iterval ad higher complexity. I the applicatio of, the problems exist i traditioal logistic chaos, such =

9 834 J. Wag ad Q. Dig distributio 5 distributio 5 distributio distributio distributio distributio Figure. The histograms of chaos at all levels x Figure. Self-correlatio performaces of chaos at all levels as arrow stable widow, the fixed-poit, etc, were effectively solved. O the basis of the compariso ad aalysis of the statistical radomess idicated by the ergodicity, self-correlatio, etc, we foud that has a better statistical radomess. Therefore, proposed i this paper would

10 Excellet Performaces of The Third-level Disturbed Chaos i The Cryptography Algorithm 835 be more suitable to be applied i the cryptography algorithm ad the spread spectrum commuicatio. Ackowledgmet. This work is supported by the Natioal Natural Sciece Foudatio of Chia (o ) ad doctor fud item of miistry of educatio (o ). REFERENCES [] H. Wag, J. D. Hu, Improved Logistic-Map Chaotic Spreadig Sequeces, Joural of Commuicatio, vol.8, pp. 7-77, 997. [2] P. A, M. Zai, Image ecryptio with chaotically coupled chaotic maps, [J]. Physica D, vol. 237, pp , 28. [3] B. Yu, L. H. Wag, Y. Q. Jia, Research ad Simulatio of Multi-User Chaotic Spread Commuicatio System Based O Improved Chaotic Spread Sequeces, [J]. Applicatio Research of Computers, vol. 4, pp. 6-65, 23. [4] C. W. Rhodes, Iterferece to digital broadbad commuicatios ad spread spectrum commuicatios, [J]. IEEE Traso Cosumer Electroics, vol. 58, o., pp. 5-22, 22. [5] R. M. Yi, J. Yua, X. M Sha, etc, Weak Keys Radomess Aalysis of Chaotic Cryptographic System, [J]. Sciece Chia, Iformatio Scieces, vol. 4, pp , 2. [6] X. Y. Yag, G. Wag, X. T. Gu, Performace Aalysis ad Applicatio Simulatio of Logistic Chaotic Sequece, [J]. Desigig Techiques of Posts ad Telecommuicatios, vol. 46, o. 2, pp. 9-22, 23. [7] D. L. Zheg, G. Zhao, G. B Xu, Logistic Mappig Digital-flow Chaotic Strage Attractor ad Its Parameter Aalysis, Joural of Uiversity of Sciece ad Techology Beijig, vol. 24, o. 3, pp , 22. [8] Q. Lu, X. H. Li, J. Li, etc., A Improved Desig of Chaotic FH Sequece, [J]. Data Acquisio & Processig, vol. 25, o., pp , 29. [9] W. Zhag, H.M. Xie, B. P. Wag, A New Algorithm of Spread Spectrum Commuicatio Based o Segmeted Logistic Chaos Spread, [J]. Computer Sciece, vol. 4, o., pp , 23. [] M. l. Zheg, Research of Wireless Sesor Network Block Cipher Based Iteger Chaos, [D]. Master s Degree Theses of South Chia Uiversity of Techology 23. [] H. J. Wag, B. B. Sog, Q. Liu, J. Pa, Q. Dig, FPGA Desig ad Applicable Aalysis of Discrete Chaotic Maps, Iteratioal Joural of Buficatio ad Chaos, vol. 24, o. 4, pp. -5, 24. [2] Y. Liu, Aalysis of Commuicatio Applicatio Based O Chaos Theory, [D]. Master s degree thesis of Xi a Electroic ad Sciece Uiversity 27 [3] G. Y. Wag, Y. Li, Reseach of Cascade Chaos ad Its Dyamic Characteristics, [J]. Chiese Joura of Physics, vol. 62, o. 2, pp. -2, 23. [4] S. X. Li, C. WANG, G. P. LI, X. M. ZHANG, L. Y. YUE. Desig ad Aalysis of New Two- Level Segmeted Logistic Chaotic Map, Joura of Northeast Normal Uiversity (Natural Sciece Editio), vol. 2, pp , 23. [5] H. Xue, H. B. Zhao, Simulatio ad Aalysis of Several Kids of Chaotic Systems With Strage Attractors, [J]. Joural of Bizhou College, vol. 26, o. 6, pp. 66-7, 2. [6] B. Che, G. H. Liu, J. Tag, et al, Research o chaotic sequece autocorrelatio by phase space method, Joural of Uiversity of Electroic Sciece arid Techology of Chia, vol. 39, o. 6, pp , 2.