Kohda, Tohru

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Title Chaotic Sequences of I.I.D Binary Random Variable with TheirApplications to CommunicationsAuthor(s)Kohda, TohruCitationChaos Memorial Symposium in Asuka : selected papersdedicated to professor Yoshisuke Ueda on the occasion of his60th birthday, p.35-44Issue Date1997URL http://hdl.handle.net/2433/24264RightType BookTextversionpublisherKyoto UniversityChaotic Sequences of I.I.D. Binary Random Variables with Their Applications to Communications Tohru KOHDA Department of  Computer Science and Communication Engineering, Kyushu University 6-1 0- 1 Hakoxaki, Higashi-ku, Fukuoka 81 2-81, Japan Abstract: The Bernoulli shift is a fundamental theoretic model of a sequence of independent and identically distributed(i.i.d.) binary random variables in probability theory, ergodic theory, information theory, and so on. We give a simple sufficient condition for a class of ergodic maps with some symmetric properties to produce a chaotic sequence of i.i.d. binary random variables. This condition is expressed in terms of binary function, which is a generalized version of the Rademacher function for the dyadic map. Applications of such a sequene to communications are also discussed briefly 1 Introduction Binary sequences[l] play an important role in modern digital communication systems, such as spread spectrum (SS) communications or cryptosystems. Although such a bi- nary sequence can be generated in various ways, linear feedback shift register (LFSR) sequences are employed in nearly all the methods [2]-[6]. It is, however, notewor- thy that a sequence of independent and identically distributed (i.i.d.) binary random variables is a typical theoretic model of pseudorandom number generators with good properties[7]-[9]. As is well known in the fields of probability theory and ergodic theory, Rademacher functions for the Bernoulli map (or the dyadic map) can produce sequences of i.i.d. random variables However, if we calculate such dynamics with the help of a computer with its necessarily limited accuracy, the period of the sequences generated from such piecewise linear maps is very short. On the other hand, it is also well known that the performance of digital communication systems depends primarily on the statistical properties of binary sequences. Neverthless, in most of various applications of chaos to communications, a number of investigators have proposed techniques to use a chaotic real-valued trajectory itself rather than its binary version, that is, analogue techniques. Such a situation motivated us to define [lo] several types of binary sequence based on a chaotic real-valued orbit generated by ergodic maps such as logistic [ll] and the Chebyshev maps [12]. Furthermore, using the Perron-Frobenius operator of some er- godic maps, we have given a simple sufficient condition for a class of binary functions to produce a sequence of i.i.d. binary random variables.[l3],[14] In this paper, we shortly review several types of method for generating chaotic sequences of i.i.d. binary random variables. Furthermore, we briefly discuss applications of such chaotic sequences to spread spectrum communications and cryptosystems. 2 Chaotic Binary Sequences Perhaps the simplest mathematical objects that can display chaotic behavior are a class of onedimensional maps [15] wn+~ = ~ ( w n )  (1) where wn = rn(wo) E I, n = 0,1,2, . . . and r (.) : I + I is a nonlinear map, where I = [d, el denotes an interval. It is known that the ensembleaverage defined by is useful in evaluating statistics of {F(T~(W))},",~ under the assumption that r(w) is mixing on I with respect to an absolutely continuous invariant (or briefly ACI) measure, denoted by f * (w) dw . Let G(w) and H(w) be any two L1 functions of bounded variation. Consider two sequences {G(T~(W))}:=~ and {H(T~(W))},",~. The 2nd-order crosscorrelation function between the two sequences from a seed w = wo is defined by where l = 0,1,2,. . .. The crosscovariance function is also defined as Note that when G = H,  these denote the autecorrelation function and auto-covariance function, respectively. If the interval I is given by I = [d,e], then the P-F operator P, of the map r is defined by [15] This operator is very useful in evaluating the correlation functions because it has the following important property: Using this property, we get The above cross-correlation function (p(')(l; G, H)) is of major importance to the investigation of statistical properties of sequences {G(~"(W))},",~ and { H ( T ~ ( W ) ) } ~ = ~ .  For several maps, such as the tent map, the logistic map, and the Chebyshev maps whose invariant density functions are known, the autecorrelation functions of real- valued sequences were already evaluated [16], [17]. In our previous study, we proposed three simple methods to obtain binary sequences from chaotic real-valued sequences { T ~ ( W ) } ~ = ~  with an ergodic map r( .)  as follows [lo]. Method-1: We define a threshold function Ot(w) as 0 for w < t Ot(w) = 1 for w 2 t and define its complementary function - Qt(w) = 1 - Ot (w). (10) Using these functions, we can obtain a binary sequence {O,(?(W))),",~, which is re- ferred to as a chaotic threshold sequence. Here define Note that pr(t) is a monotonically decreasing function of t. Method-2: We write the value of w (Iwl 5 1) in a binary representation: I w I  = O.Al(w)Az(w) A ~ ( w )  . . .  , Ai(w) E (0, 1). (13) The i-th bit Ai(w) can be expressed as 2"l = c (-llT-l {e, (w) + e-, cw)). T=l Thus we can obtain a binary sequence {Ai(wn))r=o which we call a chaotic bit sequence. Since @(w) can be regarded as a Boolean function whose variable, seed w, is not binary but real-valued, Ai (w) can be rewritten by where @ denotes modulo 2 addition. W - d  Method-3: We write the value of -E [O,l] in a binary representation: e - d  W - d  =O.Bl(w)Bz(w)...Bi(w)--., w ~ [ d , e ] ,  Bi(w)~{O,l} .  e - d  (16) The i-th bit Bi(w) can be expressed as We can obtain a binary sequence { B i ( ~ n ( ~ ) ) } ~ = o .  Note that Bi(w) can also be rewritten in the form of modulo 2 addition of threshold sequences. If the interval I = [O, 11, then Ai(w) = Bi(w). Thus each of {Ai(~n(w))},",o and {Bi(~n(w))),"_o is referred to as a chaotic bit sequence. Tausworthe [2] and Lewis & Payne [3] gave the methods to obtain a real-valued ran- dom variablk represented in a binary expansion by using shift register binary sequences. In our methods, on the contrary, we intend to get binary sequences from chaotic real- valued trajectories. This implies that our methods are inversions of Tausworthe and Lewis & Payne's generators. Now we consider a piecewise monotonic map r : [d, e] --+ [d, e] that satisfies the following properties: (i) There is a partition d = do < dl < . . . < dNT = e of [d, e] such that for each integer i = 1,. . . , N, (N, > 2) the restriction of T to the interval [di-l, di), denoted by ri (1 5 i 5 N,), is a C2 function; as well as (ii) r((di-I,&)) = (d, e), that is, ri is onto; (iii) T has a unique ACI measure denoted by f * (w)dw. The conditions for T to have a unique ACI measure are discussed in ref. [18]. For the above map, we have [15] where gi(w) = ril (w) . We now consider a class of the above piecewise monotonic maps satisfying which is referred to as the equidistributivity property [14]. Note that this class contains well known maps, such as the R-adic map, the tent map, the logistic map, and the Chebyshev map of degree k, where N, = R, 2,2, k, respectively. Thus we give the following interesting lemma [14] which is very useful in evaluating correlation functions of chaotic threshold and bit sequences. Lemma 1: For the piecewise monotonic maps satisfying eq.(19), we can get where s(w) is the signum function defined by -1 for w < 0 s(w) = 1 f o r w 2 0 .  Corollary 1: The covariance function between two chaotic threshold sequences {Ot(rn (w))}:=~ and {Otl(~n(~))),",o generated by the piecewise monotonic maps satisfying eq.(19) is evaluated as 1 ($"(e; @t, %I)) = -s((re)'(t))($2)(~; @ri(t), Qt')), N," (22) where (g2)(o; t , oil)) = p, (max[t, tl]) - ~ , ( t ) ~ ,  (tt), f 1 for e = o This corollary makes it easier to calculate the covariance function between bit sf+ quences { Ai (7" (w))},",~ (respectively, {Bi (T" (w))),",~) and {Aj (7" (w))),",~ (respec- tively, {Bj(rn(w))},",o) as follows. Remark 1: For piecewise monotonic maps satisfying the equidistributivity property eq. (19), we have (p) (e; odi, ot)) = P, ( m a  [di , tl ) - pr (di)p, (t) for e = 0 for t 2 1 which implies that there are some correlations between {Od, ( r " ( ~ ) ) } r = ~  and {Ot(r"(w))}r=o only when l = 0. Of course, if the sequences are completely independent of each other, the covariance functions should have zero value for all l. 3 Symmetric Binary Functions Now, we introduce here a new binary function. To do this, define a partition d = to < tl < . . . < taM = e of [d, e] such that and T denotes the set of symmetric thresholds {t,):$. Then we get a binary function which is referred to as a binary function with symmetric thresholds (or briefly a sym- metric binary function) [14]. Next let us restrict our attention to the map satisfying which is referred to as a symmetric property of the invariant measure. Note that such a class of maps contains well known maps, such as the R-adic map, the tent map, the logistic map, and the Chebyshev map. Remark 2: For the maps with the symmetric property of the invariant measure eq.(28), we get 1 (cT) = -. 2 (29) Furthermore, we consider a somewhat restricted class of piecewise monotonic maps satisfying eq.(19) which also satisfy the symmetric property of the map Such a class includes the tent map, the logistic map, and the Chebyshev map of even degree k. The fact that T is monotonic and onto gives The following lemma [14] plays an important role in estimating the covariance func- tions of symmetric binary sequences {CT(~n(w))),",o as shown in Corollary 2. Lemma 2: For the piecewise monotonic maps satisfying both eq. (19) and eq. (28), and their symmetric binary functions, we can get Corollary 2: Consider the piecewise monotonic maps with both eq. (19) and eq.(28). Denote two different sets of symmetric thresholds by T = {tT)zzo and TI = {t:)zfi, where Then we can obtain QgT/ - (CT)(CTt) for l = O ( $  T T ) )  = for er 1 where Remark 3: Assume Then we can get or 1 Q&, = for T # T' (44) ($')(l; CT, Cp))  = 0 for a11 t 2 0. (45) Remark 4: When M = 2i-1 and t, = (e - d)r/2i + d, we have CT (w) = Bi (w) . (46) This implies that for the piecewise monotonic maps with both eq.(19) and eq.(30), we can obtain (F(~)  (4 Bi, Bj) ) = { Yij - ( B ) ( B )  for l = 0 for e 2 1 where Note that we can easily get Qij = a for i # j, that is, ($2)(0; Bi, Bj ) )  = 0, for the maps with the uniform invariant density f *(w) = 1. On the other hand, for the maps with the nonuniform invariant densities, such as the logistic and the Chebyshev map, we can get 1 lim Qij = for i # j. i-00 (52) or j+w .Remark 5: Consider the R-adic map SR(w) defined by SR(w)=RW mod 1, R = 2 , 3 , 4 , . . . ,  W E  [O,l]. (53) For the R-adic map with even R, ($2) (e; B ~ ,  Bj))s, = o for all e. (54) Note that the symmetric binary function is a generalized version of the Rademacher function for the dyadic map [7]-[9]. 4 m-Distributivity of Chaotic Binary Sequences In the previous section, we discussed the second-order correlation functions of chaotic binary sequences. Now consider m binary functions Gi(w) (i = 1,2, .  . . , m). For m binary events gl,g2, . . . , gm (gi E (0, I), i = 1,2, .  . , m), a joint probability defined by Prob(gm, gm-1,. . . , gl) = Prob (Gm (w) = gm, Gm- (.re"-I (w)) = gm- . . . , ( ~ ~ m - l + ~ m - 2 + . . . + ~ 1  (w>) = g1),(55) l , > o ( l i i ~ m - l )  must be investigated to test the independency of sequences {Gi(~"(w))},",~ from a statistical point of view. To do this, the higher-order (the m-th order) correlation function is introduced as follows. (p(7n)(~m-~,~m-2,...,~~;Hm,Hm-i,...,Hi)) = /; H, (w)  H ~ - ~  (rem-l(w)) H ~ - ~  ( ~ ~ ~ - l +  em-2 (w)) . . . . H~ (Tem-l+!m-z+...+el (w)) f * (w)dw for all integers li 2 0, (56) where each of Hi(w) denotes an L1 real-valued function (i = 1,2, .  . . ,m).  It is, in general, difficult to evaluate such higher-order correlation functions explicitly. However, it is simplified if the following condition is satisfied. Now define a class of piecewise monotonic maps for which there is a nontrivial real-valued function H(w) satisfying which is a general version of eq. (32). Theorem 2: For any real-valued functions Hn(w) (n = 2,3,. . . , m) satisfying eq.(57), and f o r & >  1 ( n =  1 , 2 , . . . , r n - l ) ,  Note that, in the above theorem, Hl(w) need not satisfy eq. (57). Next, let cm = UoUl . . - Urn-1 be an arbitrary string of m binary digits where Un E {0,1) (0 5 n 5 rn - 1). Then there are 2" possible strings. Let d ~ )  = zlr)u?) . . . uLY1 be the r-th string with binary elements u r )  E {O,l). Furthermore, for any L1 binary function G(w), introduce a binary random variable where c (w)  = 1 - G(w) and a:) = 1 - ug). Then the probability of the event il;) in an infinite binary sequence {G(-rn(w))),",, is given by = ( p ~ m ~ ( l , l ~ ~ ~ ~ ~ l ~ ~ ~ ( ~ ~ d ~ ~ ) ~ ~ ~ ( ~ l ~ ~ ~ ) , ~ ~ ~ ~ ~ ~ - ~ ( ~ ~ d ~ ~ ) ) ) . ( 6 ~ )  - m-l We can give the following corollary [14]. Corollary 3: For any binary function B(w) satisfying eq.(57), we can easily get ~rob(z$A); B) = (B)'(l - (a))"-", where s is the number of 1 in {u$))~g~. The above corollary implies that {B(?(w))),"=, is a sequence of i.i.d. binary random variables in the sense that it can realize a Bernoulli sequence with probability (B). Note that we can get a fair Bernoulli sequence when (B) = i, that is, an m-distributed binary random sequence. 5 Applications to  Communications Spread spectrum techniques are due primarily to properties of spreading sequences (or pseudonoise (PN) sequences) [4]. Various classes of PN sequences have been pro- posed most of which are generated by LFSR sequences such as the families of the Gold sequences and of the Kasami sequences with low even-correlation values [4],[5]. Let us consider information data with different significant bits to be transmitted, such as (color) image data. In such a situation, we can assign spreading sequences of longer period to more significant bits than to less ones in order to reduce the error prob abilities of significant bits by using spreading sequences of variableperiod [I 91, [20]. Such a system becomes asynchronous. Since in asynchronous direct-sequence spread- spectrum multiple access (DSISSMA), the bit error probabilities depend primarliy on both even and odd correlation values of spreading sequences causing multipleaccess interference(MAI), we propose to use a spreading sequence of i.i.d. binary random variables whose statistical properties, such as distributions of correlation values, can be evaluated theoretically. On the other hand, stream ciphers provide probably the most important method of modern encipherment[6]. The central problem in stream cipher cryptography is the difficulty of efficiently generating unpredictable running-key sequences of binary signals from a short and random key. Although such an unpredictable sequence can be generated in various ways, LFSR sequences are employed in nea.rly all methods regardless of their possible cryptographic weaknesses. Since a chaotic sequence of i.i.d. binary random variables has several cryptographic characteristics, we can implement a stream cipher system using such a chaotic running-key sequence of i.i.d. binary random variables [21], [22]. 6 Concluding Remarks We have given simple methods to generate a sequence of i.i.d. binary random variables by means of modulo 2 addition of threshold sequences. We have also given a sufficient condition for a binary function to produce a sequence of i.i.d. binary random variables. Such a binary function is a generalized version of the Rademacher function for the dyadic map [7]-[9]. Furthermore, applications of such a sequence to communications are also briefly given. References [I] D. Knuth, The Art of Computer Programming 2, Seminumerical Algorithms, 2nd ed. Addison- Wesley, Reading, Mass, 1981. [2] R. C. Tausworthe, "Random numbers generated by linear reccurence modulo two," Mathematics of Computation, vol. 19, pp. 201-209, 1965. [3] T.  G. Lewis and W. H. Payne, "Generalized feedback shift register pseudorandom number algo- rithm?" J. ACM, vol. 20, pp. 456-468, 1973. [4] D. V. Sarwate and M. B. Pursley, "Crosscorrelation Properties of Pseudorandom and Related Sequences," Proc. IEEE, vol. 68, no. 3, pp.593-619, 1980. [5] M. K. Simon, J. K. Omura, R. A. Scholtz, and B.K.Levitt, Spread Spectrum Communications Hand book, McGraw-Hill, 1994. [6] J. L. Massey, "An Introduction to Contemporary Cryptology," Proc. IEEE, vol. 76, no. 5, pp.533- 549, May 1988. [7] M. Kac, Statistical Independence in Probability Analysis and Number Theory, The Mathematical Association of America, 1959. [8] C. M. Goldie and R. G. E. Pinch, Communication Theory, Cambridge University Press, 1991. [9] P. Billingsley, Probability and Measure. John Wiley & Sons, 1995. [lo] T. Kohda and A. Tsuneda, "Pseudonoise Sequences by Chaotic Nonlinear Maps and Their Cor- relation Properties," IEICE Trans. vol. E76-B, no. 8, pp. 855-862, 1993. [ll] S. L. Ulam and J, von Neumann, "On combination of stochastic and deterministic processes," Bull. Math. Soc. 53, p.1120, 1947. [12] R. L. Adler and T. J. Rivlin, "Ergodic and mixing properties of Chebyshev polynomials," Proc. Amer. Math. Soc. vol. 15, pp. 794-796, 1964. [13] T. Kohda and A. Tsuneda, "Explicit Evaluations of Correlation Functions of Chebyshev Binary and Bit Sequences Based on Perron-Frobenius Operator," IEICE., Trans. vol. E77-A, no. 11, pp. 1794-1800, 1994. 1141 T. Kohda and A. Tsuneda, "Statistics of Chaotic Binary Sequences", IEEE Trans. Information Theory, vo1.43, no. 1, pp. 104-112, Jan. 1997. [15] A. Lasota and M. C. Mackey, Chaos, fractals, and Noise, Springer-Verlag, 1994. [16] S. Grossmann and S. Thomae, "Invariant distributions and stationary correlation functions of one-dimensional discrete processes," Z. Naturforsch. vol. 32a, pp. 1353-1363, 1977. [17] T. Geisel and V. Fairen, "Statistical Properties of Cham in Chebyshev Maps," Physics Letters vol. 105A, no. 6, pp. 263-266, 1984. [18] A. Boyarsky and M. Scarowsky, "On a Class of Tkansformations Which Have Unique Absolutely Continuous Invariant Measures," Trans. Am. Math. Soc. vol. 255, pp. 243-262, 1979. [19] T. Kohda, A. Tsuneda, A. Osiumi and K. Ishii, "A study on pseudonoise-coded image communi- cations" Proc. of SPIE's Visual Communications and Image Processing '94, pp.874-884, 1994. [20] T. Kohda, K. Ishii, and A. Tsuneda, "Image Tkansmission Systems through CDMA Channels Using Spreading Sequences of Variable-Period", Proc. of IEEE Fourth International Symposium on Spread Spectrum Techniques & Applications, pp.781-784, 1996. [21] T. Kohda, and A. Tsuneda, "Chaotic Bit Sequences for Stream Cipher Cryptography and Their Correlation Functions", Proc. SPIE's Inter. Sympo. on Information, Communications and Com- puter Technology, Applications and Systems, vol. 2612, (Chaotic Circuits for Communication), pp.86-97,1995 [22] T. Kohda, and A. Tsuneda, "Stream Cipher Systems Based on Chaotic Binary Sequences", Proc. SCIS'96, SCIS96-11C, (1996) 

Chaotic Sequences of I.I.D Binary Random Variable with Their Applications to Communications

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Chaotic Sequences of I.I.D Binary Random Variable with Their Applications to Communications

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