Design and Analyses of some 1-D Chaotic Generators for Secure Data

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1 SETIT th International Conference: Sciences of Electronic, Tenologies of Information and Telecommnications Mar 22-26, 2009 TUISIA Design and Analyses of some 1-D Chaotic Generators for Secre Data Safwan EL ASSAD *, Hassan OURA *, Qianxe WAG *, Ina TARALOVA ** * IRREA: Instite de Reere en électroniqe et Electroteniqe de antes Atlantiqe Polyte antes, Re Christian pac BP antes Cedex 3,France safwan.elassad@niv-nantes.fr hassan.nora@niv-nantes.fr qianxe.wang@et.niv-nantes.fr ** IRCCy: Instite de reere en commnications et Cybernétiqe de antes Ina.Taralova@irccyn.ec-nantes.fr Abstract: In this paper, we stdy, design, and implement nder Matlab/Simlink some 1-D digital aos generation methods to secre data. All designed generators have the same strctre, bt sing two different non-linear fnctions (left circlate fnction, Ln x) and containing one or two cascading stages. We display the importance of the pertrbing orbit teniqe and the cascading teniqe, to avoid the dynamical degradation cased by 2 -dimensional finite space states. The first teniqe increases also the orbit cycle length. Finally, to qantify the secrity level, we perform a comparative stdy of the dynamical statistical properties: noise-like atocorrelation, cross-correlation, niformity, attractors, and IST (ational Institte of Standards and Tenology) tests obtained nder simlation. Key words: Chaotic generators, pertrbed teniqe, cascaded teniqe, IST tests. ITRODUCTIO In today's highly compterized and interconnected world, interest has been growing in the se of aos for secre commnications, and the idea of sing digital aotic systems to constrct cryptography and psedo-random coding has been extensively stdied since 1990s. The secrity of the transmitted information throgh a annel, against passive or active attacks is an international concern. The se of the aotic signals to mask sefl information and to make it nrecognizable (by modlation or encryption) by an intrder is a field of resear in fll expansion. So, we need a digital aotic signal with good cryptographic properties, s as: balance on {0, 1}; long cycle-length; δ -like atocorrelation; crosscorrelation near to zero; desired attractor; and having good tests IST. To obtain better dynamical statistical properties and avoid the dynamical degradation cased by the digital aotic systems working in a 2 -dimensional finite space, three teniqes can be sed: adding more delays, cascading mltiple aotic systems and pertrbing aotic orbit. All these teniqes are implemented and tested. In this paper, we improve the, by inclding a psedo randomly pertrbing teniqe of the aotic orbit and we stdy some new systems to generate aotic signals with desired statistical properties. After, we perform a comparative analysis of obtained simlation reslts of the dynamical signal properties before conclding. 1. Pertrbed aotic orbit teniqe To improve the dynamical signal properties, a pertrbation based algorithm is sed. Indeed, the cycle length is expanded and good statistical properties are reaed [Tao 2001], [El Assad 2008]. Considering a one dimensional aotic generator defined by: xn ( ) = Fxn [ ( 1)] 0,1 xn ( ) 0,1 n= 1,2,... (1) Here, for compting precision, ea x can be described: - 1 -

2 { } x() n = 0. () n x2()... n xi()... n x() n xi() n 0,1 i= 1,2,..., (2) The fndamental basis of the pertrbing method is the fact that no stable cycles exist, i.e. the aotic system having entered a cycle can be driven to leave the cycle immediately by a pertrbance, and will rn away from the cycle after iterations. The oose of the pertrbance is done according to the following principles: it shold have controllable long cycle length and niform distribtion; it shold not degrade the good statistical properties of aos dynamics, so the magnitde of the pertrbing signal mst be m smaller than that of the aotic signal. The signal-tonoise ratio is defined as: maximmmagnitde of aoticsignal SR=10 log( ) (3) maximmmagnitdeof pertrbingsignal A sitable candidate for the pertrbing signal generator is the maximal length LFSR becase its generated seqences have the following advantages: 1) definite cycle length (2 k - 1) (k is the degree); 2) niform distribtion; 3) delta like atocorrelation fnction; 4) easy implementation; 5) controllable maximm signal magnitde given by 2 - (2 k - 1) when sed in -precision system. The pertrbing bit for every n clock time can be generated as following: k 1 = k = k 1 k 1 Q () n Q () n g Q () n gq() n... g Q () n (4) withn = 0,1,2,... Where represents exclsive or, g 0 g 1... gk 1 are the tap coefficients of the primitive polynomial generator, and Q 0 Q 1... Qk 1 are the initial register vales of whi at least one is non zero. The pertrbance starts at n= 0, and the next ones occr periodically every Δ iterations ( Δ is a positive integer), with n= l Δ, l=1,2,, The pertrbed seqence is given by the eqation (5): F[ xi ] 1 i k xi = (5) F[ xi] Q i k 1 i Where F[ xi ( n )] represents the ith bit of Fxn [ ( )]. The pertrbance is applied on the last k bits of Fxn [ ( )]. When n l Δ, no pertrbance occrs, and then xn ( ) = Fxn [ ( 1)]. The system cycle length is given by the following relation (see appendix): ( 2 k 1) T = σ Δ (6) 2. Presentation of developed teniqes 2.1. An approa to generate aos for secre commnications has been demonstrated by Frey. The codec ses a non linear filter with finite precision ( bits) in conjnction with its inverse filter. The non linear fnction sed is the left circlate fnction sited to hardware implementation. The general eqation of Frey generator is defined by the following relation [Frey 1993], [El Assad 2006]: e () n = F { k () n [ G e ( n D) s()]} n (8) L i i i = 1 m The index in eqation (8) denotes an nsigned nmber. Also, all additions are modlo 2. These operators are assmed to be generally nonlinear operations. The generator seme consists in a non linear fnction F L (x) with a delayed feedback loop. A particlar case (3 delays) of the system driven by eqation (8), is shown in figre 1. All operations inside the loop work in the nsigned nmber representation modlo 2. The delivered aotic signal e (n) is composed by 2 qantized levels inclding the interval between[0,2 1], and having the dration of T seconds for ea ip: e = mod{ k mod{ mod[ x2 x3]}} e = mod[ k x2 x3] (9) = e x2 = e ( n 2) x3 = lcirc[ e( n 3)] = mod[2 e( n 3) s( n 1)] (10) 1 0 if e ( 3) 2 ( 1) n < sn = 1 otherwise (11) Where is the binary word length, and x 1, x 2 and x 3 are the states, namely, the otpts of the delays. Combining these eqations, we obtain: e() n = mod[ k() n e( n 1) e( n 2) 2 e( n 3) s] (12) Where σ is a positive integer. The lower bond of the system cycle length is: k ( ) T min =Δ 2 1 (7) - 2 -

3 k ( n ) e () n xtox s converter es () n 1/ 2 1 e' () n s x = e 11 1 x = e ( n 2) 12 1 x = lcirc[ e ( n 3)] = 13 1 mod[2 e ( n 3) s ] 1 1 x = e 21 2 x = e ( n 2) 22 2 x = lcirc[ e ( n 3)] = 23 2 mod[2 e ( n 3) s ] if e ( n 3) < 2 sn ( 1) = 1 otherwise 1 By combining the above eqations, we obtain: (15) (16) (17) Figre 1: Frey strctre generator nder test The inpt signal k ( n ) plays in or application the role of an additional key, whi does not allow an nathorized eavesdropper to recover the generated signal. However k ( n ) can be sed as information signal to encrypt. In order to redce the signal s mean power, and to make its amplitde independent of the nmber of levels (in fact, on ), the generated signal e (n) is, first converted into a signed signal e s (n) (The index s means «signed») in the 2 s complement in the 2 representation set, [ C 2,2 ], and then normalized by the maximm absolte vale of the qantized levels. Hence, the effective generated signal is: e ' ( n ) = q p ( t nt ) with : s n T n e1 = mod[ k e1( n 1) e1( n 2) 2 e1( n 3) s1] e2() n = mod[ k() n e1( n 1) e1( n 2) 2 e1( n 3) s1( n 1) e2( n 1) e2( n 2) 2 e2( n 3) s2] (18) k e1 ( n ) e 2 ( n ) q n < q 1 n < 2 2 and p T 1 if 0 t < T = 0 otherwise 2.2. Two Cascaded Cascading two aotic systems will let the otpt of the generator be more complex [TA 04], [WA 08]. The eqations of two cascaded are written as follows (see figre2): e1 = mod{ k mod{ 1 mod[ 2 3]}} (13) e = mod[ k x x x ] e2 = mod{ e1 mod{ x21 mod[ x22 x23]}} e = mod[ e x x x ] (14) Figre 2: Two cascaded 2.3. Generator sing logarithmic fnction as a LF The eqations of the proposed generator are written as follows (see figre 3) [WA 08]: e = mod{ k mod{ x mod[ x,1],1},1} e = mod[ k x x,1] 2 = e x2 = Ln[ e ( n 2)], 0 < e ( n 2) < 1 = e x = Ln[ e ( n 2)], 0 < e ( n 2) < 1 (19) - 3 -

4 e = floor{mod{ k e ( n 1) Ln[ e ( n 2)],1}* 2 } (20) paragraph 2, or by sing a logarithmic fnction as a non-linear fnction (see figre 3). The mod 1 operator imposes 0 < e ( n 2) < 1. Combining these eqations, we obtain: k ( n ) e 2 floor Ln Figre 4: DFT spectrm of Figre 3: Strctre sing logarithmic fnction 3. Comparative simlation reslts In order to verify reslts and compare cryptographic properties of different generators, some experiments were done. The finite compting precision is = 32 bits. The primitive polynomial is given by: x x x x 1 ( k = 16 ). The pertrbing interval is Δ= 99. Both initial conditions and control parameters are generated randomly. A large nmber of sampled vales are simlated ( samples). For example, the following parameters and initial conditions have sed: T = 5ns ; k = 0 ; x 1 (0) x 2 (0) = 400 ; x 3 (0) = and x 1 (0) = ; x 2 (0) = 402 ; 3 (0) To verify the balance property, a set of 500 seqences for ea aotic generator are compted, ea seqence containing bits. For all generators nder tests, we fond that the time domain variation of otpt signals, the spectrms (DFT), the atocorrelation fnctions and crosscorrelation fnctions are clearly noise-like. Figres 4), 5), and 6) represent the DFT spectrm, the atocorrelation fnction, the cross-correlation fnction and the attractor for the with three delays. The percentages of zeros and ones are 55% and 45% respectively. In figres 7), 8) and 9) we show the DFT spectrm, the atocorrelation fnction the crosscorrelation fnction and the attractor of pertrbed Frey map, whi are more random signal compared with reslts of figres 5), 6). The percentages of zeros and ones are # 50%. Roghly, we obtain the same reslts of cascaded two by inclding to with three delays, the pertrbation teniqe described in Figre 5: Ato and Cross correlation of Figre 6: Attractor of - 4 -

5 test. From this table, we can conclde that Frey map with two cascaded stage, generator with logarithmic fnction and pertrbed are sitable for sing in crypto-systems based aos. Figre 7: DFT spectrm of pertrbed 4. Conclsion In a crypto-system, the se of a good aotic generator, with desirable dynamical statistical properties, is very important. In this paper, we have developed and implemented nder Matlab/Simlink for 1-D dimensional aotic generators. A comparative analysis of performances sing standard criteria between these generators proves the efficiency of the pertrbed and cascading teniqes. As prospect of this work, we are working on a generator strctre whose cycle length is as big as needed. ACKOWLEDGMET The athors wish to thank the Fédération AtlanSTIC CRS FR2819. This work was spported by the AtlanSTIC resear clster. Figre 8: Ato & crosss correlation of pertrbed Figre 9: Attractor of pertrbed Table 1 shows the obtained reslts of the IST statistical tests [RUK 01] of stdied generators. Table 1: IST tests Statistical tests with three Pertrbed Two cascaded delays and Ln fnction Parameters P-val % P-val % P-val % REFERECES [RUK 01] A. Rkhin, J. Soto, J. evatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, (2001). A Statistical Test Site For Random and Psedorandom mber Generators FOR CRYPTOGRAPHIIC APPLIICATIIOS. IST Special Pblication (with revisions dated May 15, 2001). [FRE 93] D. R. Frey Chaotic digital encoding: an approa to secre commnications, IEEE Transactions on Circits and Systems II, vol. 40, pp , [TAO 98] S. Tao, W. Rilli, Y. Yixn, Pertrbance based algorithm to expand cycle length of aotic key stream, Electronics Letters, 34(9):pp , [ASS 06] S. El Assad, C. Vladean, Digital aotic codec for DS-CDMA Commnication Systems, Lebanese Science Jornal, vol. 7, o. 2, [ASS 06] S. El Assad, Commnications mériqes: Teniqes Avancées, cors 5ème année, à Polyte antes, [WA 08] Q. Wang Performances of some Digital Chaotic Generators nder IST tests, Master2R SEGE, Polyte antes,, [TA 04] W. P. Tang, H. K. Kwan, «Chaotic commnications sing nonlinear transfor-pairs», ISCAS-2004, pp, V , We se 100 samples of 10 6 bit seqences for ea - 5 -