Improving Middle Square Method RNG Using Chaotic Map

Transcription

1 Applied Mathematics, 20, 2, doi:0.4236/am Published Olie April 20 ( Improvig Middle Square Method RNG Usig Chaotic Map Abstract Hamed Rahimov, Majid Babaie, Hassa Hassaabadi 2 Departmet of Computer Egieerig, Shahrood Uiversity of Techology, Shahrood, Ira 2 Physics Departmet, Shahrood Uiversity of Techology, Shahrood, Ira hrahimov@shahroodut.ac.ir, hrahimov@gmail.com Received Jauary 23, 20; revised March 0, 20; accepted March 2, 20 Oe of the classic approaches i PRNGs is the middle square method i which with a simple mathematical model geeratig pseudoradom umbers i high speed ad miimum correlatio betwee output umbers. Despite these uique characteristics, the method cotais weakesses that a broader applicatio of this algorithm will face. I this paper is studied middle square method ad the a logistic chaotic map is itroduced with its specific features ad its improved weakesses via usig these characteristics. Fially the NIST tests suite s are preseted, i order to detect the specific characteristics expected from truly radom sequeces. Keywords: Middle Square Method (MSM), Radom Number Geerator, Logistic Map, NIST Tests Suite. Itroductio Nowadays, the world is coected together by etworks. The mai problem i the preset etworks is the security protocols. This is due to growig iterest i the use of digital chaotic systems offerig the possibility to support the security of cryptographic algorithms for data ecrypttio, like those preseted i the etworkig security protocols [,2]. The best ways to icrease security i moder commuicatio methods is usig the cryptography algorithms. Most of these ew algorithms use chaotic maps as pseudoradom umber geerators (PRNGs) [3,4] to obtai a biary stream icludig symmetric ecryptio algorithm [5]. PRNGs are very importat i several fields like statistical studies, simulatios, cryptography ad solvig specific mathematical problems [6-8]. They may be based o upredictable physical resources such as physical pheomeo or o mathematical algorithms like Halto algorithm [6]. However, truly radom umbers (i.e. physical pheomeo) are ot gaied by a specific algorithm [8,9] as they are used i the ecryptio systems [9]. So we should make use of ay real implemetatio some simple fuctios for RNG istead of TRNGs [0,]. A PRNG should be checked by certai statistical tests which it must pass all statistical tests that are restricted to polyomial time i the size of the seed ad frequecy of 0 s ad s i the specific sequeces [2]. I this paper we have preseted a chaotic RNG based o middle square method to prepare a mathematical model for geeratig radom umbers. Our model is the tested by the best kow statistical method, i.e. the NIST test suite. Oe of the fudametal subjects i geeratig radom umbers is the Uiform distributio [] i the iterval [0, ] (e.g. geeratig real umbers like: or ). Geeratig i this iterval eables us to ivestigate our practical purposes for itroducig a efficiet ecryptio method. The reaso is that the samples from the other distributios are derived usig the Uiform Distributio []. So it is importat that our proposed geerator has the uiform distributio. This part itroduces a brief history of some PRNG algorithms, The Va der Corput [6], Halto [6] ad Sobol [6] sequeces are three basic mathematical algorithms for PRNGs. These are based o low discrepacy method with which uique properties are used i the effective areas such as army idustry. Nevertheless, they have some disadvatages i high dimesios, a example beig the geerated umbers i 20 2 dimesios for Halto sequece due to multidimesioal clusterig [6] (as show i Figure ). I order to clarify the purpose of the proposed algorithm, oe ca see some of methods i Refereces [3-27]. With this backgroud, it is uderstood that i the recet years the PRNG s algorithm developed by some

2 H. RAHIMOV ET AL. 483 Figure. Halto sequeces multidimesioal clusterig: Dimesios Figure 2. MSM radom sequece (With core = 8439). physical algorithm such as chaos theory ad produced radom umbers rapidly ad with less correlatio [3-27]. This paper ivestigates a efficiet method to produce radom umbers with the combiatio of middle Square Method (MSM) ad chaotic logistic map. Hece i the ext sectio, we describe the radom umber s applicatio i the cryptography algorithm. 2. Middle Square Method Middle Square Method is a oe of the best methods of geeratig pseudoradom umbers. But i the some practical situatios it is ot a good method, sice its period is usually very short (as show i the Figure 2 ad Figure 3). This mathematical algorithm was described by Joh vo Neuma i 949 [28]. I this method for geeratig a sequece of te digit pseudoradom umbers, i the first step a four digits startig value is created ad squared (e.g. A A = C) so that a eight digits umber is produced. The by a simple mathematical model geerated the middle four digits of the result ad it would be the ext umber i the sequece (e.g. D = C mod 0 /2 ) ad retured as the result. By repeatig this process we are able to geerate more radom umbers. Oe of the mai problems occurs whe the middle /2 digits (i this example, four digits) are all zero ad the geerator yields zero outputs forever. For testig this algorithm we take some radom umbers ad describe the weakesses i our examples which would describe a = 8439 for first example (as show i Figure 2) ad a = 6395 for secod example (as show i Figure 3) [28]. Sequece oe, with core umber (8439) accordig with Figure 2. After geeratig 24 radom umbers we have a ulimited loop geeratig 0. Bearig i mid the limited radom umbers geerated by this method, i this paper proposed a ovel Figure 3. MSM radom sequece (With core = 6395). chaotic system based o logistic map. Most of these discrete chaotic cryptosystem algorithms use a chaotic map as PRNGs to geerate a keystream which is used for ecryptio of a plaitext message to produce a ciphertext [5]. The secret key of such systems costitutes the iitial values ad/or the system parameters [5]. I this paper, a ovel sequece geerator is proposed which is based o a sigle oe-dimesioal logistic map. Sequece three, with core umbers (6395); accordig with Figure 3 after geeratig 26 radom umbers we have a loop with legth 602 (i.e. where umbers: 27 ad 89, we have geerated 0.25, so the legth of the loop is 602). This paper is structured as follows. I Sectio 3, the properties of the logistic map are discussed. I Sectio 4, a complete descriptio of the chaotic middle square method is preseted. The statistical properties of these sequeces are discussed i Sectio 5 ad fially the re- sult of NIST test preseted i Table. 3. Chaotic Logistic Map As we kow, the logistic map is the oe of the simplest

3 484 H. RAHIMOV ET AL. ad most studied oliear that was itroduced as a demographic model. The logistic map is give by that is show i Equatio (): f x x r x x () Where x is the state variable, which geerates values i the iterval [0, ] ad r is called system parameter which ca have ay value betwee oe ad four [0,29]. There is a o-trivial fixed poit where 3.4 r 4 which is see i Figure 4. Fially for r 4, we observe that chaos values are geerated i the complete rage of [0, ]. I this paper, the logistic map has applicatio of Table. Result of NIST tests suite for proposed RNG. Test P-value Pass rate Frequecy Block-Frequecy CuSums-forward CuSums-backward Ras Log ru Rak FFT Overlappig templates Uiversal Approximate etropy Serial Serial Lier complexity the form: 4. Proposed Chaotic PRNG Middle square method has some big problems [28] that were metioed i Sectio 3 with the otice of these problems i MSM ad cosider to uique properties of chaotic logistic map i this sectio preseted a chaotic solutio for solvig these problems. Whe efficiet values are chose for umbers i the core that have o loop with the specific legth, ext we ca correct the ulimited loop (whe PRNG joust 0 geerated i sequece) by exchagig oe of the core s umbers [28]. C A A D C% t %0 E C D Result E Numt Result 0 if ( Numt 0) the { x xr x B x x x } Assume the logistic chaotic map with r = 4 i x 0 = 0.6 Geerates a radom umber ad replaces it with a umber i A s place ad cotiues geeratig radom umbers without ay ulimited loop ad the loop with a particular legth. These problems ca be solved by implemetatio of chaotic iteratio that is geerated i parallel. Middle square method algorithm described i Equatio (2) also see below, havig the umber (i.e. A). (2) 5. NIST Statistical Tests Suite Figure 4. Bifurcatio plot of the Logistic map i x 0, r 3.4,4. The NIST tests suite is a statistical package ivolvig 5 tests ad is based o hypothesis testig. Also The NIST tests suite focus o a variety of differet types of o-radomess. I the preset case, the tests cosist determiig whether or ot a specific sequece of zeroes ad oes is radom [8]. The theoretical backgroud of its distributio is based o ull hypothesis which is determied by mathematical methods ad correspodig probability value (P-value). The P-value is the probability that a perfect RNG for each test would produce a sequece less radom tha the sequece that was tested, give the kid of o-radomess assessed by the test. If the P-value for a test is determied to be equal zero, that meas the sequece ap-

4 H. RAHIMOV ET AL. 485 pears to be exactly o-radom ad a P-value to be, the the sequece has perfect radomess [8]. 6. Coclusios I this paper we have proposed a pseudoradom umber geerator (PRNG) based o the combiatio of chaotic logistic map ad Middle Square Method, the chaotic system iterated idepedetly startig from iitial coditios could help to geerate appropriate values; we have also tested the geerated sequeces usig the NIST tests to detect the uique characteristics expected from truly radom bit sequeces. The results of statistical testig are reliable so has bee used as a part of ecryptio system to geerate secret key ad have a efficiet algorithm. Fially the result of their statistical tests preseted i the Table. 7. Refereces [] Y. Hu, X. Liao, K. W. Wog ad Q. Zhou, A True Radom Number Geerator Based o Mouse Movemet ad Chaotic Cryptography, Chaos, Solitos ad Fractals, Vol. 40, No. 5, 2009, pp doi:0.06/j.chaos [2] P. L. Ecuyer ad J. Grager-Piché, Combied Geerators with Compoets from Differet Families, Mathematics ad Computers i Simulatio, Vol. 62, No. 3, 2003, pp doi:0.06/s (02) [3] M. Orlov, Optimized Radom Number Geeratio i a Iterval, Iformatio Processig Letters, Vol. 09, No. 3, 2009, pp doi:0.06/j.ipl [4] Q. Zhou, X. Liao, K. W. Wog, Y. Hua ad D. Xiao, True Radom Number Geerator Based o Mouse Movemet ad Chaotic Hash Fuctio, Iformatio Scieces, Vol. 79, No. 9, 2009, pp doi:0.06/j.is [5] K. Nakao, K. Kawakami ad K. Shigemoto, RSA Ecryptio ad Decryptio Usig the Redudat Number System o the FPGA, IEEE Iteratioal Symposium o Parallel & Distributed Processig, Rome, May 2009, pp. -8. [6] I. Krykova, Evaluatig of Path-Depedet Securities with Low Discrepacy Methods, Master of Sciece Thesis, Worce-Ster Polytechic Istitute, [7] T. Stojaovski ad L. C. Kocarev, Chaos-Based Radom Number Geerators, IEEE Trasactios o Circuits ad Systems, Vol. 48, No. 3, 200, pp [8] K. H. Tsoi, K. H. Leug ad P. H. W. Leog, High Performace Physical Radom Number Geerator, IET Proceedig Computers & Digital Techiques, Vol., No. 4, 2007, pp doi:0.049/iet-cdt: [9] M. I. Youssef, M. Zahara, A. E. Emam ad M. A. Elghay, Image Ecryptio Usig Pseudo Radom Number ad Chaotic Sequece Geerators, Radio Sciece Coferece, New Cairo, 7-9 March 2009, pp. -5. [0] C. Pellicer-Lostao ad R. Lopez-Ruiz, Pseudo-Radom Bit Geeratio Based o 2D Chaotic Maps of Logistic Type ad Its Applicatios i Chaotic Cryptography, Joural of Computatioal Sciece ad Its Applicatios, Vol. 5073, 2008, pp [] X.-J. Tog, M.-G. Cui ad W. Jiag, The Productio Algorithm of Pseudo-Radom Number Geerator Based o Compoud No-Liear Chaos System, Proceedigs of the Iteratioal Coferece o Itelliget Iformatio Hidig ad Multimedia Sigal Processig, Califoria, 8-20 December 2006, pp doi:0.09/iih-msp [2] Q. Wag, C. Guyeux ad J. M. Bahi, A Novel Pseudo- Radom Number Geerator Based o Discrete Chaotic Iteratios, First Iteratioal Coferece o Evolvig Iteret, Caes/La Bocca, August 2009, pp doi:0.09/internet [3] H. P. Lü, S. H. Wag ad G. Hu, Pseudo-Radom Number Geerator Based o Coupled Map Lattices, Iteratioal Joural of Moder Physics B, Vol. 8, No. 7-9, 2004, pp [4] X. M. Li, H. B. She ad X. L. Ya, Characteristic Aalysis of a Chaotic Radom Number Geerator Usig Piece-Wise-Liear Map, Joural of Electroics ad Iformatio Techology, Vol. 27, No. 6, 2005, pp [5] J. Liu, Desig of a Chaotic Radom Sequece ad Its Applicatio, Computer Egieerig, Vol. 3, No. 8, 2005, pp [6] Y. Wag, H. She ad X. Ya, Desig of a Chaotic Radom Number Geerator, Chiese Joural of Semicoductors, Vol. 26, No. 2, 2005, pp [7] L. Wag, F. P. Wag ad Z. J. Wag, Novel Chaos- Based Pseudo-Radom Number Geerator, Acta Physica Siica, Vol. 55, No. 8, 2006, pp [8] M. Adrecut, Logistic Map as a Radom Number Geerator, Iteratioal Joural of Moder Physics B, Vol. 2, No. 9, 998, pp doi:0.42/s x [9] J. A. Gozalez ad R. Pio, Radom Number Geerator Based o Upredictable Chaotic Fuctios, Computer Physics Commuicatios, Vol. 20, No. 2-3, 999, pp doi:0.06/s (99) [20] L. Kocarev ad G. Jakimoski, Pseudo-Radom Bits Geerated by Chaotic Maps, IEEE Trasactios o Circuits ad Systems, Vol. 50, No., 2003, pp doi:0.09/tcsi [2] S. M. Fu, Z. Y. Che ad Y. A. Zhou, Chaos Based Radom Number Geerators, Computer Research ad Developmet, Vol. 4, No. 4, 2004, pp [22] S. Ergu ad S. Ozoguz, Truly Radom Number Geerators Based o a No-autoomous Chaotic Oscillator, AEU-Iteratioal Joural Electroics & Commuicatios, Vol. 6, No. 4, 2007, pp doi:0.06/j.aeue [23] V. V. Kolesov, R. V. Belyaev ad G. M. Voroov, A Digital Radom-Number Geerator Based o the Chaotic

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