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The Graph Structure of the Generalized Discrete Arnold’s Cat Map Chengqing Li, Kai Tan, Bingbing Feng, Jinhu Lu¨
Abstract—Chaotic dynamics is an important source for gener- [17]–[19]. In [20], Cat map is used as an example to define ating pseudorandom binary sequences (PRNS). Much efforts have a microscopic entropy of chaotic systems. The nice properties been devoted to obtaining period distribution of the generalized of Cat map demonstrated in the infinite-precision domains discrete Arnold’s Cat map in various domains using all kinds of theoretical methods, including Hensel’s lifting approach. Diago- seemingly support that it is widely used in many cryptographic nalizing the transform matrix of the map, this paper gives the applications, e.g. chaotic cryptography [21], image encryption explicit formulation of any iteration of the generalized Cat map. [22], image privacy protection [23], [24], hashing scheme Then, its real graph (cycle) structure in any binary arithmetic [25], PRNG [11], [26], random perturbation [27], designing domain is disclosed. The subtle rules on how the cycles (itself unpredictable path flying robot [28]. and its distribution) change with the arithmetic precision e are elaborately investigated and proved. The regular and beautiful Recently, the dynamics and randomness of digital chaos are patterns of Cat map demonstrated in a computer adopting fixed- investigated from the perspective of functional graphs (state- point arithmetics are rigorously proved and experimentally ver- mapping networks) [29]. In [30], how the structures of Logistic ified. The results will facilitate research on dynamics of variants map and Tent map change with the implementation precision of the Cap map in any domain and its effective application in e is theoretically proved. Some properties on period of a cryptography. In addition, the used methodology can be used to evaluate randomness of PRNS generated by iterating any other variant of Logistic map over Galois ring Z3e are presented maps. [31], [32]. In [33], the phase space of cat map is divided Index Terms—cycle structure, chaotic cryptography, fixed- into some uniform Ulam cells, and the associated directed point arithmetic, generalized Cat map, period distribution, complex network is built with respect to mapping relationship PRNS, pseudorandom number sequence. between every pair of cells. Then, the average path length of the network is used to measure the underlying dynamics of Cat map. In [34], the elements measuring phase space structures I.INTRODUCTION of Cat map, fixed points, periodic orbits and manifolds (stable ERIOD and cycle distribution of chaotic systems are or unstable), are detected with Lagrangian descriptors. In [35], P fundamental characteristics measuring their dynamics and the functional graph of general linear maps over finite fields is function, and supporting their practical values [1], [2]. As studied with various network parameters, e.g. the number of the most popular application form, various digitized chaotic cycles and the average of the pre-period (transient) length. To systems were constructed or enhanced as a source of produc- quickly calculate the maximal transient length, fixed points and ing random number sequences: Tent map [3], Logistic map periodic limit cycles of the functional graph of digital chaotic [4], [5], Cat map [6], Chebyshev map of even degree [7], maps, a fast period search algorithm using a tree structure is piecewise linear map [8], and Chua’s attractor [9]. Among designed in [36]. them, Arnold’s Cat map The original Cat map (1) can be attributed to the general f(x, y) = (x + y, x + 2y) mod 1 (1) matrix form f(x) = (Φ · x) mod N, (2) is one of the most famous chaotic maps, named after Vladimir arXiv:1712.07905v5 [nlin.CD] 24 Sep 2019 Arnold, who heuristically demonstrated its stretching (mixing) where N is a positive integer, x is a vector of size n × 1, effects using an image of a Cat in [10, Fig. 1.17]. Attracted and Φ is a matrix of size n × n. The determinant of the by the simple form but complex dynamics of Arnold’s Cat transform matrix Φ in Eq. (2) is one, so the original Cat map map, it is adopted as a hot research object in various domains: is area-preserving. Keeping such fundamental characteristic of quadratic field [1], two-dimensional torus [11]–[13], [14], Arnold’s Cat map unchanged, it can be generalized or extended finite-precision digital computer [15], [16], quantum computer via various strategies: changing the scope (domain) of the This work was supported by the National Natural Science Foundation of elements in Φ [37]; extending the transform matrix to 2-D, China (no. 61772447, 61532020). 3-D and even any higher dimension [6], [38]; modifying the C. Li is with College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, Hunan, China modulo N [39]; altering the domain of some parameters or ([email protected]). variables [40]. K. Tan and B. Feng are with College of Information Engineering, Xiangtan Among all kinds of generalizations of Cat map, the one University, Xiangtan 411105, Hunan, China. J. Lu¨ is with School of Automation Science and Electrical Engineering, in 2-D integer domain received most intensive attentions due Beihang University, Beijing 100083, China to its direct application on permuting position of elements of 1549-8328 c 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. Wednesday 25th September, 2019 (02:15) 2 IEEE TRANSACTIONS
√ image data, which can be represented as where λ+ = (1 + 5)/2. Under specific conditions on prime decomposition forms of N or the parity of T , the two bounds x x x f n = n+1 = C · n mod N, (3) in Eq. (5) are further optimized in [41]. y y y n n+1 n When N is a power of two, as for any (p, q), [16] gives where possible representation form of the period of Cat map (3) 1 p C = , (4) over Galois ring Z2e , shown in Property 1. Furthermore, the q 1 + p · q relationship between T and the number of different Cat maps + possessing the period, NT , is precisely derived: xn, yn ∈ ZN , and p, q, N ∈ Z . In this paper, we refer to the generalized Cat map (3) as Cat NT = map for simplicity. In [41], the upper and lower bounds of the 1 if T = 1; period of Cat map (3) with (p, q) = (1, 1) are theoretically 3 if T = 2; derived. In [42], the corresponding properties of Cat map (3) 2e+1 + 12 if T = 4; with (p, q, N) satisfying some constrains are further disclosed. In [15], [16], [43], [44], F. Chen systematically analyzed the 2e−1 + 2e if T = 6; (6) precise period distribution of Cat map (3) with any parameters. e+k−2 2k−2 k 2 + 3 · 2 if T = 2 , k ∈ {3, 4, ··· , e − 1}; The whole analyses are divided into three parts according to 2e−2 e 2 if T = 2 ; influences on algebraic properties of ( , +, ·) imposed by ZN 2e+k−1 if T = 3 · 2k, k ∈ {0, 2, 3, ··· , e − 2}, N: a Galois field when N is a prime [43]; a Galois ring when N is a power of a prime [15], [16]; a commutative where e ≥ 4. ring when N is a common composite [44]. According to the analysis methods adopted, the second case is further divided Property 1. The representation form of T is determined by into two sub-cases N = pe and N = 2e, where p is a parity of p and q: e ( prime larger than or equal to 3 and is an integer. From the 2k, if 2 | p or 2 | q; viewpoint of real applications in digital devices, the case of T = 0 (7) 3 · 2k , if 2 - p and 2 - q. Galois ring Z2e is of most importance since it is isomorphic to the set of numbers represented by e-bit fixed-point arithmetic where k ∈ {0, 1, ··· , e}, k0 ∈ {0, 1, ··· , e − 2}. format with operations defined in the standard for arithmetic of computer. When e = 3, The period of a map over a given domain is the least 1 if T = 1; common multiple of the periods of all points in the do- main. Confusing periods of two different objects causes some 3 if T = 2; e−1 misunderstanding on impact of the knowledge about period 2 if T = 3; NT = distribution of Cat map in some references like [42]. What’s 2e+1 + 12 if T = 4; worse, the local properties of Arnold’s Cat map are omitted. e−1 e 2 + 2 if T = 6; Diagonalizing the transform matrix of Cat map with its eigen- 22e−2 if T = 8, matrix, this paper derives the explicit representation of any iteration of Cat map. Then, the evolution properties of the which cannot be presented as the general form (6) as [16, internal structure of Cat map (3) with incremental increase of Table III], e.g. 22e−2 6= 2e+k−2 + 3 · 22k−2 when e = k = 3. e are rigorously proved, accompanying by some convincing From Eq. (6), one can see that there are (1 + 3 + 2e−1 + experimental results. 2e+1 + 12 + 2e−1 + 2e) = 2e+2 + 16 generalized Arnold’s The rest of this paper is organized as follows. Section II maps whose periods are not larger than 6, which is a huge gives previous works on deriving the period distribution of number for ordinary digital computer, where e ≥ 32. Cat map. Section III presents some properties on structure of The generating function of the sequence generated by Cat map. Application of the obtained results are discussed in iterating Cat map (3) over Z2e from initial point (x0, y0) can Sec. IV. The last section concludes the paper. be represented as g (t) X(t) = x II.THEPREVIOUSWORKSONTHEPERIODOF CAT MAP f(t) To make the analysis on the (overall and local) structure and g (t) of Cat map complete, the previous related elegant results are Y (t) = y , briefly reviewed in this section. f(t) 1 1 0 1 2 When (p, q) = (1, 1), C = [ 1 2 ] = [ 1 1 ] , the period where problem of Cat map (3) can be transformed as the divisibility " # " # " # " # g (t) −1 − p · q p x x properties of Fibonacci numbers [41]. Then, the known the- x = · 0 · t + 0 , orems about Fibonacci numbers are used to obtain the upper gy(t) q −1 y0 y0 and lower bounds of the period of Cat map (3), i.e. and 2 e logλ+ (N) < T ≤ 3N, (5) f(t) = t − ((pq + 2) mod 2 ) · t + 1. LI et al. 3
0 TABLE I: The conditions of (p, q) and the number of their possible cases, NT , corresponding to a given T . 0 T p q NT 1 0 0 1 p mod 2e = 2e−1 q mod 2e−1 = 0 2 2 q mod 2e = 0 p mod 2e = 2e−1 1 3 p ≡ 1 mod 2 q ≡ p−1(2e − 3) mod 2e 2e−1 p ≡ 1 mod 2 q ≡ p−1(2e − 2) mod 2e 2e−1 p ≡ 0 mod 2, p 6≡ 0 mod 4 q ≡ (p/2)−1(2e−1 − 1) mod 2e−1 2e−1 p ≡ 1 mod 2 q ≡ p−1(2e−1 − 2) mod 2e 2e−1 4 p ≡ 0 mod 2 q ≡ (p/2)−1(2e−1 − 2) mod 2e 2e−1 p mod 2e−1 = 2e−2 q mod 2e−2 = 0 8 q mod 2e−1 = 0 p mod 2e−1 = 2e−2 4 q ≡ p−1(2e−1 − 3) mod 2e 2e−1 6 p ≡ 1 mod 2 q ≡ p−1(2e−1 − 1) mod 2e 2e−1 q ≡ p−1(2e − 1) mod 2e 2e−1 p ≡ 1 mod 2 q ≡ p−1(2e−k+1l − 2) mod 2e, l ≡ 1 mod 2, l ∈ [1, 2k−1 − 1] 2e+k−3 2k, p ≡ 0 mod 2 q ≡ (p/2)−1(2e−k+1l − 2) mod 2e, l ≡ 1 mod 2, l ∈ [1, 2k−1 − 1] 2e+k−3 k ∈ {3, 4, . . . , e − 1} p mod 2e−k+1 = 2e−k q mod 2e−k = 0 22k−1 p mod 2e−k+1 = 0 q mod 2e−k+1 = 2e−k 22k−2 p ≡ 1 mod 2 q ≡ 0 mod 4 22e−3 2e p ≡ 0 mod 4 q ≡ 1 mod 2 22e−3 3 · 2k, q ≡ p−1(2e−kl − 3) mod 2e, l ≡ 1 mod 2, l ∈ [1, 2k] 2e+k−2 p ≡ 1 mod 2 k ∈ {2, 3, . . . , e − 2} q ≡ p−1(2e−k+1l − 1) mod 2e, l ≡ 1 mod 2, l ∈ [1, 2k−1 − 1] 2e+k−2
Referring to Property 2, when p and q are not both even, the the functional graph of Cat map (3) change with the arithmetic period of Cat map is equal to the period of f(t). So the period precision e, let problem of Arnold’s Cat map becomes that of a decomposition e part of its generation function. First, the number of distinct zn,e = xn,e + (yn,e · 2 ), (8) Cat maps possessing a specific period over Z2[t] is counted. where xn,e and yn,e denote xn and yn of Cat map (3) with Then, the analysis is incrementally extended to e [t] using the Z2 N = 2e, respectively. Hensel’s lifting approach. As for any given value of the period As a typical example, we depicted the functional graphs of of Arnold’s Cat map, all possible values of the corresponding 4 Cat map (3) with (p, q) = (1, 1) in four domains { e } in (p, q) are listed in Table I 1. Z2 e=1 Fig. 1, where the number inside each circle (node) is zn,e in Property 2. As for Cat map (3) implemented over (Z2e , +, ·), Fe. From Fig. 1, one can observe some general properties of there is one point in the domain, whose period is a multiple functional graphs of Cat map (3). Especially, there are only of the period of any other points. cycles, no any transient. The properties on permutation are concluded in Properties 3, 4.
III.THESTRUCTUREOF CAT MAP OVER (Z2e , +, · ) Property 3. Cat map (3) defines a bijective mapping on the set (0, 1, 2, ··· ,N 2 − 1). First, some intuitive properties of Cat map over (Z2e , +, · ) are presented. Then, some general properties of Cat map over Proof. As Cat map (3) is area-preserving on its domain, (ZN , +, · ) and (Z2e , +, · ) are given, respectively. Finally, 2 it defines a bijective mapping on ZN , which is further the regular graph structures of Cat map over ( 2e , +, · ) are Z transformed into a bijective mapping on ZN 2 by conversion disclosed with the properties of two parameters of Cat map’s function (8). explicit presentation matrix. Property 4. As for a given N, any node of functional graph of Cat map (3) belongs one and only one cycle, a set of nodes A. Properties of funtional graph of Cat map over ( e , +, · ) Z2 such that Cat map (3) iteratively map them one to the other Functional graph of Cat map (3) can provide direct per- in turn. spective on its structure. The associate functional graph Fe can be built as follows: the N 2 possible states are viewed as Proof. Referring to [45, Theorem 5.1.1], the set 2 (0, 1, 2, ··· ,N 2 − 1) N nodes; the node corresponding to x1 = (x1, y1) is directly is divided into some disjoint subsets linked to the other one corresponding to x2 = (x2, y2) if and such that Cat map (3) is a cycle on each subset. only if x2 = f(x1) [30]. To facilitate visualization as a 1-D network data, every 2-D vector in Cat map (3) is transformed As the period of a Cat map in a domain is the least common multiple of the periods of its cycles, the functional graph of by a bijective function zn = xn + (yn · N). To describe how a Cat map possessing a large period may be composed of a 1 To facilitate reference of readers, we re-summarized the results in [16] great number of cycles of very small periods. The whole graph in a concise and straightforward form. shown in Fig. 1d) is composed of 16 cycles of period 12, 10 4 IEEE TRANSACTIONS
2 16 48 20 32 0 52 22 50 54 36 18 34 38 6 4 2 9 0 8 4 46 30 24 44 3 63 33 41 51 29 49 11 0 1 10 7 27 7 13 37 57 55 47 53 3 11 5 14 62 56 12 2 3 6 12 1 8 58 25 10 15 13 14 17 15 1 45 60 23 35 59 43 21 9 5 19 28 31 39 40 26 61 42 a) b) c) 36 32 172 108 96 72 128 192 64 4 134 230 66 46 6 238 130 162 198 106 0 204 136 196 76 200 12 8 140 132 68 110 142 166 74 102 14 42 138 226 206 164 160 44 236 224
126 149 121 235 37 216 25 129 197 65 112 56 151 83 186 122 228 98 168 2 170 232 78 194 38 209 62 231 189 144 33 5 221 34 10 70 104 174 202 190 89 53 239 41 16 85 13 100 234 40 73 205 184 240 219 31 250 58 29 254 227 241 88 45 137 145
179 139 244 185 84 19 167 52 199 82 147 109 158 222 35 39 233 229 177 245 243 63 92 148 7 255 117 188 87 156 124 171 18 75 97 95 119 15 60 253 28 223 163 252 67 146215 105 94 30 47 43 237 225 125 57 183 123 20 220 131 59 49 116 155 212 180 175 210 207 101 27
54 217 127 135 90 191 213 141 107 176 152 23 201 69 214 22 79 195 242 178 113 181 161 173 21 246 3 187115 154 1 153 248 111 91 208 118 157 51 11 26 251 17 9 103 120 80 211 77 193 150 86 203 71 50 114 61 249 165 169 81 182 143 247 55 218 133 93 48 99 159 24 d) Fig. 1: Functional graphs of generalized Arnold’s Cat maps in Z2e where (p, q) = (1, 1): a) e = 1; b) e = 2; c) e = 3; d) e = 4.
Assign (an0 , bn0 ) with one element in set (15), one can obtain 3 0 0 2 0 1 the corresponding cycle in Fe+1 with the steps given in 2 1 3 1 3 2 Property 6. Then, the other element in set (15) can be assigned to (a , b ) if it does not ever exist in the set in Eq. (14) a) b) c) d) n0 n0 corresponding to every assigned value of (a , b ). Every Fig. 2: Four possible functional graphs of Cat map (3) with n0 n0 cycle corresponding to different (a , b ) can be generated N = 2: a) p and q are both even; b) p is even, and q is odd; n0 n0 c) p is odd, q is even; d) p and q are both odd. in the same way. Property 5. If the differences between inputs of Cat map (3) cycles of period 6, 1 cycle of period 3, and 1 self-connected with N = 2e and that of Cat map (3) with N = 2e+1 satisfy cycle. " # " # We found that there exists strong evolution relationship xn,e+1 − xn,e an e e = · 2 , (10) between Fe and Fe+1. A node zn,e = xn,e + yn,e2 in Fe yn,e+1 − yn,e bn is evoluted to one has e e e+1 zn,e+1 = (xn,e + an2 ) + (yn,e + bn2 )2 " # "" # " # " ## an+1 1 p an kx e e 2e+1 = · + mod 2, (11) = zn,e + (an2 + yn,e2 + bn2 ), (9) bn+1 q 1 + p · q bn ky
0 e 0 e where an, bn ∈ {0, 1}. The relationship between iterated node an, bn ∈ {0, 1}, kx = bkx/2 c, ky = bky/2 c, and of zn,e in Fe and the corresponding evoluted one in Fe+1 " 0 # " # " # is described in Property 5. Furthermore, the associated cycle kx 1 p xn,e 0 = · . (12) is expanded to up to four cycles as presented in Property 6. ky q 1 + p · q yn,e LI et al. 5
0 0 0 0 15 14 13 12 0 3 0 0 3 1 11 3 10 2 14 10 6 2 0 10 2 8 3 0 2 2 0 1 1 2 11 10 9 8 3 0 0 3 9 1 8 0 12 8 4 0 11 3 6 4 2 1 2 1 2 2 3 3 0 7 6 5 4 15 13 7 5 15 7 11 3 7 13 15 5 1 1 2 2 1 1 3 3 1 1 2 3 2 2 2 3 3 1 3 1 0 2 1 3 3 2 3 0 3 0 2 0 2 1 0 1 1 3 2 1 0 14 12 6 4 13 5 9 1 9 1 14 12 2 3 1 0 1 3 0 2 1 0 2 0) 2) 8) 10) 3 a) b) c) d) 0 3 2 1 0 3 2 1 0 2 4 7 0 2 4 5 Fig. 4: Mapping relationship between (an +2bn) and (an+1 + 10 8 11 9 10 8 11 9 10 8 13 12 10 8 15 12 12 13 4 7 12 15 4 5 1 11 5 14 1 11 13 6 2bn+1) in Eq. (11) with (kx + 2ky) shown beside the arrow:
15 14 5 6 13 14 7 6 9 3 6 15 9 3 14 7 a) p and q are both even; b) p is even, and q is odd; c) p is 1) 3) 9) 11) odd, q is even; d) p and q are both odd.
12 8 14 6 12 8 14 6 0 8 4 6 0 8 4 6 Depending on the number of candidates for (an0 , bn0 ) and 0 4 10 2 0 4 10 2 10 2 14 12 10 2 14 12 the corresponding cardinality in Eq. (14), a cycle of length 3 7 1 13 3 7 1 13 9 7 1 15 5 9 1 7 Tc in Fe is expanded to five possible cases in Fe+1: 1) one 15 11 5 9 15 11 5 9 13 11 5 3 11 15 13 3 cycles of length Tc and one cycle of length 3Tc, e.g. the self- 4) 12) 6) 14) connected cycle in Fig. 1a), “0 → 0”, is evoluted to two cycles 8 2 9 2 8 14 0 10 0 8 1 3 0 8 4 2 10 0 10 7 in Fig. 1b), “0 → 0” and “(0 + 2 ) = 2 → (0 + 2 ) = 8 → 10 7 6 5 12 15 2 3 9 13 4 1 3 13 6 9 1 3 3 11 5 4 1 12 (0 + 2 + 2 ) = 10 → 2”; 2) twos cycles of length Tc and 6 12 1 7 11 3 12 7 1 11 14 5 6 15 15 13 14 14 15 4 5 11 13 9 one cycle of length 2Tc, e.g. the cycle “0 → 0” in Fig. 2c) is 5) 7) 13) 15) expanded to three cycles in the same SMN: “0 → 0”; “2 → Fig. 3: All possible functional graphs of Cat map (3) with N = 2”; “8 → 10 → 8”; 3) four cycles of length Tc, e.g. the cycle 22, where the subfigure with caption “i)” is corresponding to “1 → 13 → 12 → 3 → 7 → 4 → 1” in Fig. 1b) is expanded (p, q) satisfying i = p mod 4 + (q mod 4) · 4. to four cycles of the same length shown in the lower left side of Fig. 1c). 4) two cycles of length 2Tc, e.g. the cycle in Fig. 1a), “1 → 3 → 2 → 1” is evoluted to the other two 1 3 3 Proof. According to the linearity of Cat map (3), one can get cycles in Fig. 1b), “1 → (3 + 2 + 2 ) = 13 → (2 + 2 + 2) = 12 → (1+21) = 3 → (3+21 +2) = 7 → (2+21) = 4 → 1”; " # 3 1 3 1 xn+1,e+1 − xn+1,e “(1+2 ) = 9 → (3+2 +2 +2) = 15 → (2+2 +2) = 6 → (1 + 21 + 23) = 11 → (3 + 21) = 5 → (2 + 21 + 23 + 2) = yn+1,e+1 − xn+1,e "" #" # " ## 14 → 9”; 5) one cycle of length 4Tc, e.g. the cycle “1 → 1” 1 p x − x k = n,e+1 n,e + 2e x mod 2e+1.(13) in Fig. 2c) is expanded to “1 → 9 → 3 → 11 → 1” in q 1 + p · q yn,e+1 − yn,e ky the subfigure with caption “9)” in Fig. 3. In all, all the fives
0 0 m possible cases can be found in Fig. 1, 2, 3. As k·a ≡ k·a (mod m) if and only if a ≡ a (mod gcd(m,k) ) and an+1,e, bn+1,e ∈ {0, 1}, the property can be proved by As shown in Property 6, any cycle of Fe+1 is incrementally putting condition (10) into equation (13) and dividing its both expanded from a cycle of F1. So, the number of cycles of sides and the modulo by 2e. a given length in Fe+1 has some relationship with that of Tc−1 the corresponding length in Fe, which is determined by the Property 6. Given a cycle Ze = {zn,e}n=0 = {(xn,e, Tc−1 control parameters p, q. Moreover, as shown in Property 7, yn,e)}n=0 in Fe and its any point zn0,e, one has that the Fe is isomorphic to a part of Fe+1, which can be verified in cycle to which zn0,e+1 belongs in Fe+1 is Fig. 1. Z = {z )}n0+kTc−1 , e+1 n,e+1 n=n0
i e Tc where Property 7. Any cycle {(C · X) mod 2 }i=1 in Fe and the i e+1 Tc corresponding cycle {(C ·(2X)) mod 2 }i=1 in Fe+1 com- k = #{(an0 , bn0 ), (an0+Tc , bn0+Tc ), (an0+2Tc , bn0+2Tc ), pose two isomorphic groups with respect to their respective
(an0+3Tc , bn0+3Tc )}, (14) operators. 0 n +3T z = z 0 for n ≥ T , n = n mod T , {(a , b )} 0 c n,e n ,e c c n n n=n0 are generated by iterating Eq. (11) for n = n0 ∼ n0 + 3Tc, Proof. As for any point X in Fe, define a multiplication oper- i e Tc and #(·) returns the ardinality of a set. ation ◦ for any two elements of set G = {(C ·X) mod 2 }i=1, i1+i2 e i1 e g1 ◦ g2 = (C · X) mod 2 , where g1 = (C · X) mod 2 , Proof. Given a node in a cycle, (kx, ky) in Eq. (11) is fixed, i2 e g2 = (C ·X) mod 2 . The set G is closed with respect to the so Eq. (11) defines a bijective mapping on set operator ◦. Point (CTc · X) mod 2e = (C0 · X) mod 2e = X {(0, 0), (0, 1), (1, 0), (1, 1)} (15) is the identity element. Multiplication of any three matrices satisfy the associative law. As for any element g1, there is an n z {z } Tc−i1 e as shown in Fig. 4. So, n,e+1 may fall in set j,e+1 j=n0 inverse element (C · X) mod 2 . So, the non-empty set when and only when (n − n0) mod Tc = 0 and n > n0, i.e. G composes a group with respect to the operator. Referring to the given cycle is went through one more times. the elementary properties of congruences summarized in [46, 6 IEEE TRANSACTIONS
P.61], equation Theorem 1. The n-th iteration of Cat map matrix (4) satisfies " # " # " 1 A−2 # n Gn − Hn p · Hn i x0 e x0 C = 2 2 , (16) C · mod 2 = q · H 1 G + A−2 H y0 y0 n 2 n 2 n where ( holds if and only if G = ( A+B )n + ( A−B )n, n 2 2 (17) " # " # H = 1 (( A+B )n − ( A−B )n), 2x0 2x0 n B 2 2 Ci · mod 2e+1 = . √ 2y0 2y0 B = A2 − 4 and A = p · q + 2.
0 i e+1 Tc Proof. First, one can calculate the characteristic polynomial So G = {C ·(2X) mod 2 }i=1 also composes a group with 0 0 i1+i2 of Cat map matrix (4) as respect to operator ◦ˆ, where g1◦ˆg2 = (C · (2X)) mod e+1 0 i1 e+1 0 i2 " # 2 , g1 = (C · (2X)) mod 2 , g2 = (C · (2X)) mod 1 − λ p 2e+1. Therefor, the two groups are isomorphic with respect to |C − λI| = det q p · q + 1 − λ bijective map y = (2X) mod 2e+1. = λ2 − (p · q + 2)λ + 1 The period distribution of cycles in Fe follows a power- = 0. law distribution of fixed exponent one when e is sufficiently large. The number of cycles of any length is monotonously Solving the above equation, one can obtain two characteristic increased to a constant with respect to e, which is shown in roots of Cat map matrix: Table II, where the dashline marked the case corresponding to A + B λ = , the threshold value. 1 2 In [16], it is assumed that e ≥ 3 “because the cases when A − B λ2 = . e = 1 and e = 2 are trivial”. On the contrary, the structure of 2 functional graph of Cat map (3) with e = 1, shown in Fig. 2, Setting λ in (C − λI) · X = 0 as λ1 and λ2 separately, plays a fundamental role for that with e ≥ 3, e.g. the cycles A−2+B | the corresponding eigenvector ξλ1 = [1, 2p ] and ξλ2 = of length triple of 3 in Fe (if there exist) are generated by the A−2−B | [1, 2p ] can be obtained, which means cycle of length 3 in F1. C · P = P · Λ, (18)
TABLE II: The number of cycles of period Tc in Fe with where P = (ξλ1 , ξλ2 ) and (p, q) = (9, 14). " # λ1 0 NT ,e Tc c 20 21 22 23 24 25 26 27 Λ = . e 0 λ2 1 2 1 0 0 0 0 0 0 From Eq. (18), one has 2 2 1 3 0 0 0 0 0 −1 3 2 1 15 0 0 0 0 0 C = P · Λ · P , (19) 4 2 1 63 0 0 0 0 0 where " # 5 2 1 255 0 0 0 0 0 − A−2−B p 6 2 1 1023 0 0 0 0 0 −1 2B B P = A−2+B p . 7 2 1 4095 0 0 0 0 0 2B − B 8 2 1 16383 0 0 0 0 0 Finally, one can get 9 2 1 16383 24576 0 0 0 0 n −1 n 10 2 1 16383 24576 49152 0 0 0 C = (P · Λ · P ) 11 2 1 16383 24576 49152 98304 0 0 = P · Λn · P−1 12 2 1 16383 24576 49152 98304 196608 0 " 1 A−2 # 13 2 1 16383 24576 49152 98304 196608 393216 2 Gn − 2 Hn pHn = 1 A−2 , qHn 2 Gn + 2 Hn
where Gn and Hn are defined as Eq. (17). B. Properties on iterating Cat map over (ZN , +, · ) Proposition 1. The least period of Cat map (3) over Diagonalizing the transform matrix of Cat map (3) with its (ZN , +, · ) is T if and only if T is the minimum possible eigenmatrix, the explicit representation of n-th iteration of the value of n satisfying map can be obtained as Theorem 1, which serves as basis of p · Hn ≡ 0 mod N, the analysis of this paper. The necessary and sufficient condition for the least period q · Hn ≡ 0 mod N, of Cat map (3) over (ZN , +, · ) is given in Proposition 1. (20) Considering the even parity of Gn, the condition can be 1 1 Gn − p · q · Hn ≡ 1 mod N, simplified as Corollary 1. Based on the property of Hn in 2 2 Lemma 2, the inverse of the n-th iteration of Cat map is 1 1 Gn + p · q · Hn ≡ 1 mod N, obtained as shown in Proposition 2. 2 2 LI et al. 7
∞ Lemma 2. Sequence {Hn}n=1 satisfies m−1 Proof. If the period of Cat map (3) over (ZN , +, · ) is T , T Y C · X ≡ X mod N exists for any X. Setting X = [1, 0]| and H2m·s = Hs · G2j ·s, (24) X = [0, 1]| in order, one can get j=0 " # where m and s are positive integers. 1 0 Cn ≡ mod N (21) 0 1 Proof. This Lemma is proved via mathematical induction on m. When m = 1, when n = T . Incorporating Eq. (16) into the above equation, 2s 2s! one can assure than Eq. (20) exists when n = T . As T is the 1 A + B A − B H2s = − least period, T is the minimum possible value of n satisfying B 2 2 T Eq. (20). The condition is therefore necessary. If is the = Hs · Gs. (25) minimum possible value of n satisfying Eq. (20), Eq. (21) holds, which means that T is a period of Cat map (3) over Now, assume the lemma is true for any m in Eq. (24) less k m = k (ZN , +, · ). As Eq. (21) does no exist for any n < T , T is the than . When , least period of Cat map (3) over ( , +, · ). So the sufficient k k ! ZN 1 A + B 2 ·s A − B 2 ·s part of the proposition is proved. H k = − 2 ·s B 2 2 Lemma 1. For any positive integer m, the parity of G2ms is k−1 k−1 ! 1 A + B 2 ·s A − B 2 ·s the same as that of Gs, and = − 1 B 2 2 G2m·s ≡ 1 mod 2 k−1 k−1 ! 2 A + B 2 ·s A − B 2 ·s · + if Gs is even, where s is a given positive integer. 2 2
Proof. Referring to Eq. (17), one has = H2k−1·s · G2k−1·s. (26) m m A + B 2 ·s A − B 2 ·s The above induction completes the proof of the lemma. G2m·s = + 2 2 Proposition 2. The inverse of the n-th iteration of Cat map m−1 m−1 !2 A + B 2 ·s A − B 2 ·s matrix = + " # 2 2 1 A−2 −n 2 Gn + 2 Hn −p · Hn C = 1 A−2 . (27) 2m−1·s −q · H G − H A2 − B2 n 2 n 2 n −2 4 2 Proof. Referring to Eq. (22) and Lemma 2, one has = (G2m−1·s) − 2. (22) " # 2 1 A−2 When m = 1, G2s = (Gs) − 2. So the parity of G2s is 2n 2 G2n − 2 H2n p · H2n C = 1 A−2 the same as that of Gs no matter Gs is even or odd. In case q · H2n 2 G2n + 2 H2n G is even, 1 G = 1 (G )2 − 1 ≡ 1 mod 2. Proceed by " # s 2 2s 2 s 1 G2 − 1 − A−2 H G p · H G induction on m and assume that the lemma hold for any m = 2 n 2 n n n n q · H G 1 G2 − 1 + A−2 H G less than a positive integer k > 1. When m = k, G2k·s = n n 2 n 2 n n 2 " # (G2k−1·s) − 2, which means that the parity of G2k·s is the n −1 0 same as that of G2k−1·s no matter G2k−1·s is even or odd. = Gn · C + . 1 1 2 0 −1 If Gs is even, 2 G2k·s = 2 (G2k−1·s) − 1 ≡ 1 mod 2 also holds. So, the 2n-th iteration of Cat map matrix (4) satisfies n 2n n n Corollary 1. If Gn is even, condition (20) is equivalent to Gn · C − C = C · (Gn ·I2 − C )
= I2. p · Hn ≡ 0 mod N, Substituting Cn in the above equation with Eq. (16), one can q · H ≡ 0 mod N, n get Eq. (27). 1 (23) p · q · Hn ≡ 0 mod N, 2 1 C. Properties on iterating Cat map over (Z2eˆ, +, · ) Gn ≡ 1 mod N. 2 In this sub-section, how the graph structure of Cat map changes with the binary implementation precision are dis- 1 1 1 Proof. If Gn is even, 2 Gn is an integer. As 2 Gn ± 2 p · q · Hn closed. To study the change process with the incremental 1 1 is an integer, 2 Gn − 2 p · q · Hn is also an integer. So, one can increase of the precision e from one, let eˆ denote the given 1 1 get 2 p · q · Hn ≡ 0 mod N and 2 Gn ≡ 1 mod N from the implementation precision instead, which is the upper bound last two congruences in condition (20). of e. 8 IEEE TRANSACTIONS
Using Proposition 1 and Lemma 3 on the greatest common One can verify that divisor of three integers, the necessary and sufficient condition ( for the least period of Cat map (3) over (Z2e , +, · ) is min(ep, eq) if ep + eq > 0; min(min(ep, eq), ep+eq−1) = simplified as Proposition 3. Then, Lemmas 35, 38, and 6 −1 if ep + eq = 0. describe how the two parameters of the least period of Cat 1 If min(e , e ) < e, one has min(min(e , e ), e) = map, 2 Gn and Hn, change with the implementation precision. p q p q min(e , e ). So, h can be calculated as Eq. (30). Proposition 3. The least period of Cat map (3) over p q e (Z2e , +, · ) is T if and only if T is the minimum value of Lemma 3. For any integers a, b, n, one has n satisfying 1 n n ng G ≡ 1 mod 2e, (28) gcd (gcd(d , a), gcd(d , b)) = d , 2 n e−he Hn ≡ 0 mod 2 , (29) where where n = max{x | gcd(a, b) mod dn ≡ 0 mod dx}, g −1 if ep + eq = 0; d is a prime number, lcm and gcd denote the operator solving he = min(ep, eq) if e > min(ep, eq), ep + eq 6= 0; (30) the least common multiple and greatest common divisor of two e if e ≤ min(e , e ), p q numbers, respectively. x ep = max{x | p ≡ 0 mod 2 }, eq = max{x | q ≡ 0 mod x x Proof. Let ad = max{x | a ≡ 0 mod d }, bd = max{x | b ≡ 2 }, e ≥ 1, and p, q ∈ Z2eˆ. x 0 mod d }, so min{n, ad, bd} = max{x | gcd(a, b) mod Proof. Setting N = 2e in Eq. (23), its last congruence dn ≡ 0 mod dx}. Then, one has becomes Eq. (28). Referring to Corollary 1, one can see n n that this proposition can be proved by demonstrating that gcd (gcd(d , a), gcd(d , b)) Eq. (29) is equivalent to the first three congruences in Eq. (23). = gcd gcd(dn, dad ), gcd(dn, dbd ) Combining the first two congruences in Eq. (23), one has = dmin{min{n,ad},min{n,bd}} e−he,1 min{n,a ,b } Hn ≡ 0 mod 2 , (31) = d d d
e e ng e−he,1 2 2 = d . where 2 = lcm gcd(2e,p) , gcd(2e,q) . Referring to e Lemma 3, one can get he,1 = max{x | gcd(p, q) mod 2 ≡ 0 mod 2x} as 2e 2e 2e Lemma 4. Given an integer e > 1, if m and s satisfy lcm , = . gcd(2e, p) gcd(2e, q) gcd(gcd(2e, p), gcd(2e, q)) 1 e G m ≡ 1 mod 2 , The third congruence in Eq. (23) is equivalent to 2 ·s 2 (34) e e 1 ( 2 ·gcd(2 ,2) e+1 0 mod e if ep + eq = 0; G2m·s 6≡ 1 mod 2 , gcd(2 ,p·q) 2 Hn ≡ 2e (32) 0 mod gcd(2e, 1 p·q) if ep + eq > 0. 2 one has Combining Eq. (31) and Eq. (32), one can get he = 1 e+2l−1 G2m+l·s ≡ 1 mod 2 , min(he,1, he,2) to assure that Eq. (29) is equivalent to the 2 first three congruences in Eq. (23), where 1 e+2l G m+l ≡ 1 mod 2 , (35) 2 2 ·s ( min(e, ep + eq) − 1 if ep + eq = 0; 1 e+2l+1 he,2 = (33) G2m+l·s 6≡ 1 mod 2 , min(e, ep + eq − 1) if ep + eq > 0. 2
From the definition of he,1 and he,2, one has where s and l are positive integers and m is a non-negative ( integer. min(ep, eq) if e > min(ep, eq); he,1 = Proof. This lemma is proved via mathematical induction on e if e ≤ min(ep, eq), 1 e l. From condition (34), one can get 2 G2m·s = 1 + ae · 2 , and ( where ae is an odd integer. Then, one has ep + eq − 1 if e ≥ ep + eq; he,2 = 1 1 2 e if e < ep + eq. G m+1 = (G m − 2) 2 2 ·s 2 2 ·s So, 1 = ((2 + a · 2e+1)2 − 2) 2 e min(min(ep, eq), ep + eq − 1) if e ≥ ep + eq; = 2e+2 · a · (a · 2e−1 + 1) + 1. (36) min(min(e , e ), e) if min(e , e ) < e e e h = p q p q e e−1 < ep + eq; As ae · (ae · 2 + 1) is odd, condition (35) exists for e if e ≤ min(ep, eq). l = 1. Assume that condition (35) hold for l = k, namely LI et al. 9
1 e+2k 2 G2m+k·s = 1+ae+2k ·2 , where ae+2k is an odd integer. where T1 is the least period of Cat map (3) over (Z2, +, ·). When l = k + 1, one has Proof. Depending on the parity of p, q, the proof is divided 1 1 2 G m+k+1 = (G m+k − 2) into the following three cases: 2 2 ·s 2 2 ·s • When p, q are both odd: 1 e+2k+1 2 = ((2 + ae+2k · 2 ) − 2) " # 2 1 0 = a2 · 22e+4k+1 + a · 2e+2k+2 + 1 C ≡ mod 2. e+2k e+2k 0 1 2 e+2k−1 e+2k+2 = (ae+2k · 2 + ae+2k) · 2 + 1. One can calculate T1 = 3. From Eq. (17), one has 2 e+2k−1 As a · 2 + ae+2k is an odd integer, condition (35) e+2k 3 3 also hold for l = k + 1. A + B A − B G3 = + 1 2 2 Lemma 5. If there is an odd integer a1 satisfying G2m·s = 2 2A3 + 6A · B2 2 · a1 + 1, namely = 8 1 3 G2m·s ≡ 1 mod 2, = A − 3A. 2 (37) 1 2 As A = p · q + 2 is odd, G is even. G2m·s 6≡ 1 mod 2 , T1 2 1 1 • When only p or q is odd: C ≡ [ 0 1 ] mod 2 if p is odd; one has ≡ [ 1 0 ] mod 2 q T = 2 1 C 1 1 if is odd. In either sub-case, 1 eg,0 G2m+1·s ≡ 1 mod 2 , and A is even. As 2 (38) 1 2 2 eg,0+1 A + B A − B G2m+1·s 6≡ 1 mod 2 , G = + 2 2 2 2 x where eg,0 = 3 + max{x | (a1 + 1) ≡ 0 mod 2 }. = A2 − 2, 1 3 Proof. From Eq. (36), one has G2m+1·s = 2 ·a1·(a1+1)+1. 2 one has GT is even. Then, condition (38) can be derived. 1 • When p, q are both even: T1 = 1. So GT1 = A is also Lemma 6. If Gs is even, even. ( e Substituting A = p · q + 2 into GT in each above case, one H2m·s ≡ 0 mod 2 , 1 e+1 (39) can obtain Eq. (45). H2m·s 6≡ 0 mod 2 , Property 8. The length of any cycle of Cat map (3) imple- one has k eˆ−1 ( e+l mented over (Z2eˆ, +, ·) comes from set {1} ∪ {2 · T1}k=0, H m+l ≡ 0 mod 2 , (40) 2 ·s where e+l+1 H2m+l·s 6≡ 0 mod 2 , (41) 3 if p and q are odd; where l is a positive integer. T1 = 2 if only p or q is odd; Proof. From Lemma 1, one has 1 if p and q are even,
l−1 and eˆ ≥ 2. Y l G j ≡ 0 mod 2 , 2 ·s (42) Proof. As j=0 " # " # " # l−1 1 p x x Y l+1 · mod 2e = (46) G j 6≡ 0 mod 2 , 2 ·s (43) q 1 + p · q y y j=0 if and only if since 2 | G2j ·s and 4 - G2j ·s for any j ∈ {0, 2, ··· , l − 1}. ( e From Lemma 2, one can get q · x ≡ 0 mod 2 , p · y ≡ 0 mod 2e, H2m+l·s = H2l·(2m·s) l−1 1 is the length of the cycles whose nodes satisfying condi- Y = H2m·s · G2j ·s. (44) tion (46). From Fig. 2, one can see that j=0 " #T1 " # " # 1 p x x So, Eq. (40) and inequality (41) can be obtained by combining · mod 2 = (47) Eq. (39) with Eq. (42) and inequality (43), respectively. q 1 + p · q y y
Proposition 4. For any p, q, GT1 is even, and exists for any x, y ∈ Z2 if Eq. (46) does not hold for e = 1 and T1 6= 1. From Fig. 3, one has 1 p · q(p · q + 3)2 if p and q are odd; 1 2 " #n " # " # G − 1 = p · q( 1 p · q + 2) if p or q is odd; (45) 1 p x x 2 T1 2 · mod 22 = (48) 1 2 p · q if p and q are even, q 1 + p · q y y 10 IEEE TRANSACTIONS
always holds for any x, y ∈ Z22 when n = 2 · T1 if Eq. (48) Proof. When e = 1, one has does not hold for n = T1. As G2T1 and H2T1 are both even, 1 G ≡ 1 mod 2 " #n " # " # 2 T1 1 p x e x 1−h1 · mod 2 = HT ≡ 0 mod 2 , (52) q 1 + p · q y y 1 from Proposition 3. From Eq. (52), one can get es,h, the can be presented as the equivalent form minimal number of e satisfying e " # " 1 # HT1 ≡ 0 mod 2 , x 2 · p · y − p · q · x Gn − 1 e · 2 mod 2 = 0 (49) H 6≡ 0 mod 2e+1, y 2 · q · x + p · q · y 1 H T1 2 n e ≥ 1 − h1. when n = 2T . If Eq. (49) does not hold for n = 2T and 1 1 From Lemma 5, one can get the minimum positive number of e ≥ 3, n = 22T is the least number of n satisfying Eq. (49) 1 e satisfying (See Lemmas 4 and 6). Referring to Lemmas 1 and 2, G2mT 1 1 m e and H2 T1 are both even for any positive integer m. So the m G2 0 ·T1 ≡ 1 mod 2 , equivalent form (49) can be reserved for any possible values 2 (53) 1 e+1 of n. Iteratively repeat the above process, the length of cycle m G2 0 ·T1 6≡ 1 mod 2 , k 2 n = 2 ·T1 can be obtained, where k ranges from 0 to eˆ−1. es,g, by increasing e from 1, where
If p and q are both odd, HT1 = (p·q+1)·(p·q+3) ≡ 0 mod ( 1 2 2 1 if 2 · GT1 6≡ 1 mod 2 ; 2 . As shown in Fig. 1, the length of the maximum cycle of Cat m0 = 0 otherwise. map over Z22 is T1. So, the length of the maximum cycle is 3·2eˆ−2 if eˆ ≥ 3. In addition, Property 2 is a direct consequence Referring to Proposition 4, GT1 is even. So, one can get of Property 8. 1 es,g +2·x G2m0+x·T ≡ 1 mod 2 Proposition 5. As for any p, q, Hn is even, where n is the 2 1 m0+x+es,h length of a cycle of Cat map (3) larger than one. m +x H2 0 ·T1 ≡ 0 mod 2 x Proof. Referring to the definition of HT1 in Eq. (17), by referring to Lemmas 6 and 4, where is a non-negative h integer. Note that e is monotonically increasing with respect (p · q + 1) · (p · q + 3) if p and q are odd; e hˆ e ≥ min(e , e ) to and fixed as e when p q (See Eq. (30)), H = where T1 p · q + 2 if only p or q is odd; ( ˆ −1 if ep + eq = 0; 1 if p and q are even he = (50) min(ep, eq) otherwise. can be calculated as the proof of Proposition 4. Set ˆ • When p or q is even: H2 = H1 · G1 = p · q + 2 is even. es = (es,h + m0 + he) + x0, (54) • When p and q are both odd: H3 = (p · q + 1) · (p · q + 3) is even. one has es ≥ min(ep, eq) from Eq. (50), 1 Referring to Property 8, Hn = H2m·s, where es,g +2x0 G m +x ≡ 1 mod 2 2 0 0 ·T1 ( 2 1 2 if p and q are even; es,g +2x0+1 G m +x 6≡ 1 mod 2 s = 2 0 0 ·T1 T1 otherwise. 2 and From Lemma 2, one can get Hn is even. es−hˆe m +x H2 0 0 ·T1 ≡ 0 mod 2
es+1−hˆe H2m0+x0 ·T 6≡ 0 mod 2 D. Disclosing the regular graph structure of Cat map 1 1 where With increase of e, G n and H n will reach the 2 2 T1 2 T1 ( ˆ ˆ balancing condition given in Proposition 3. As shown the proof (es,h + m0 + he) − es,g if es,g < es,h + m0 + he; x0 = in Theorem 2, the explicit presentation of the threshold value 0 otherwise. of e, es, is obtained. When e ≥ es, the period of Cat map (55) double for every increase of e by one. As for Table II, es = 8 Referring to Lemma 4, one has (The dashlined row with tiny gap). 1 es,g +2x0+2l G m +x +l ≡ 1 mod 2 . (56) 2 2 0 0 ·T1 Theorem 2. There exists a threshold value of e, es, satisfying Combing Lemma 6, T = 2l · T (51) e+l e 1 es+l m +x +l G2 0 0 ·T1 ≡ 1 mod 2 when e ≥ es, where Te is the period of Cat map over 2 es+l−hˆe ( e , +, · ), and l is a non-negative integer. m +x +l Z2 H2 0 0 ·T1 ≡ 0 mod 2 (57) LI et al. 11
0 TABLE III: The threshold values es, es under various combinations of (p, q). 0 es(e ) q s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 p 1 2(2) 3(3) 3(3) 1(1) 3(3) 4(4) 4(4) 1(1) 2(2) 3(3) 3(3) 1(1) 4(4) 5(5) 5(5) 1(1) 2 3(3) 2(1) 4(4) 1(1) 3(3) 2(1) 5(5) 1(1) 3(3) 2(1) 4(4) 1(1) 3(3) 2(1) 6(6) 1(1) 3 3(3) 4(4) 2(2) 1(1) 5(5) 3(3) 3(3) 1(1) 3(3) 6(6) 2(2) 1(1) 4(4) 3(3) 4(4) 1(1) 4 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 2(2) 5 3(3) 3(3) 5(5) 1(1) 2(2) 6(6) 3(3) 1(1) 4(4) 3(3) 4(4) 1(1) 2(2) 4(4) 3(3) 1(1) 6 4(4) 2(1) 3(3) 1(1) 6(6) 2(1) 3(3) 2(1) 4(4) 2(1) 3(3) 1(1) 5(5) 2(1) 3(3) 1(1) 7 4(4) 5(5) 3(3) 1(1) 3(3) 3(3) 2(2) 1(1) 7(7) 4(4) 4(4) 1(1) 3(3) 3(3) 2(2) 1(1) 8 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 3(3) 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 3(3) 9 2(2) 3(3) 3(3) 1(1) 4(4) 4(4) 7(7) 1(1) 2(2) 3(3) 3(3) 1(1) 3(3) 8(8) 4(4) 1(1) 10 3(3) 2(1) 6(6) 1(1) 3(3) 2(1) 4(4) 1(1) 3(3) 2(1) 5(5) 1(1) 3(3) 2(1) 4(4) 1(1) 11 3(3) 4(4) 2(2) 1(1) 4(4) 3(3) 4(4) 1(1) 3(3) 5(5) 2(2) 1(1) 5(5) 3(3) 3(3) 1(1) 12 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 2(2) 13 4(4) 3(3) 4(4) 1(1) 2(2) 5(5) 3(3) 1(1) 3(3) 3(3) 5(5) 1(1) 2(2) 4(4) 3(3) 1(1) 14 5(5) 2(1) 3(3) 1(1) 4(4) 2(1) 3(3) 1(1) 8(8) 2(1) 3(3) 1(1) 4(4) 2(1) 3(3) 1(1) 15 5(5) 6(6) 4(4) 1(1) 3(3) 3(3) 2(2) 1(1) 4(4) 4(4) 3(3) 1(1) 3(3) 3(3) 2(2) 1(1) 16 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 3(3) 1(1) 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 4(4)
as es,g + 2x0 + 2l ≥ es + l for any non-negative integer l. from Eq. (49). Setting n = Tc, Hence, by Proposition 3, " # " # 2Tc x e+1 x l m0+x0+l C · mod 2 = . (63) Te+l = 2 · Te = 2 · T1 y y when e ≥ es. Referring to Lemma 2, and Gn is even, From the proof of Theorem 2, one can see that the value " # 2n x e+1 of es in Eq. (54) is conservatively estimated to satisfy the C · mod 2 required conditions (The balancing conditions may be obtained y ˆ " # when he is still not approach he). In practice, there exists x 0 n e+1 another real threshold value of e, e ≤ e , satisfying = (Gn · C − I2) mod 2 s s y l T 0 = 2 · T 0 , es+l es " # " #! n x x e+1 = GnC − mod 2 . (64) which is verified by Table III. y y Theorem 3. When T > T , c es Therefore, 2N = N , (58) " # Tc,e 2Tc,e+1 a · 2e C2n · mod 2e+1 where N is the number of cycles with period T of Cat e Tc,e c b · 2 map (3) over (Z2e , +, ·). " e# " e#! n a · 2 a · 2 e+1 Proof. As for any point (x, y) of a cycle with the least period = GnC − mod 2 b · 2e b · 2e T in F , one has c e " # " # " # a · 2e x x e+1 Tc e = mod 2 , (65) C · mod 2 = (59) b · 2e y y " # " # where a, b ∈ {0, 1}. Combing Eq. (63) and Eq. (65) with Tc x x 2 e C · mod 2 6= (60) n = Tc, one can get y y " e# " e# 2Tc x + a · 2 e+1 x + a · 2 by referring to Property 2 and Table I. Referring to Eq. (56) C · e mod 2 = e . and Eq. (57), one has y + b · 2 y + b · 2
( Tc eg,n = es,g + 2x0 + 2l, Setting n = 2 in the left-hand side of Eq. (62), one has (61) ˆ " # " # eh,n = es − he + l, x x CTc · mod 2e+1 6= 1 x where eg,n = max{x | 2 Gn ≡ 1 mod 2 }, eh,n = max{x | y y x l Hn ≡ 0 mod 2 }, n = 2 · Te , l is a non-negative integer. s from Eq. (60) as Tc ≥ T . Combing the above inequalities So, one can get 2 es with Eq. (65), one has " # " 1 # x 2 · p · y − p · q · x 2 G2n − 1 e+1 " e# " e# · mod 2 = 0 (62) T x + a · 2 x + a · 2 y q · x + 1 p · q · y H C c · mod 2e+1 6= . 2 2n y + b · 2e y + b · 2e 12 IEEE TRANSACTIONS
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11 12 m 2 9 10 log
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5 6 19 17 19 3 4 15 17 13 15 1 11 2 13 9 11 7 e 9 19 5 0 7 17 15 13 11 3 5 9 7 1 18 16 5 3 14 12 3 1 10 8 log Tc 6 4 1 2 2 0 a) b)
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15 16
13 14
12 11 10 9 8 7 6 5 19 4 19 17 17 3 15 15 13 2 13 1 11 11 9 9 7 0 7 5 5 18 16 3 18 16 14 3 14 12 10 12 10 8 6 1 8 6 4 1 4 2 0 2 0 c) d) Fig. 5: The cycle distribution of Cat map (3) over Z2e , e = 1 ∼ 19: a) (p, q) = (7, 8); b) (p, q) = (6, 7); c) (p, q) = (5, 7); d) (p, q) = (12, 14).
e e So, 2Tc is the least period of (x + a · 2 , y + b · 2 ) in Fe+1 Lemma 8. When e > e0, any point (x, y) in a cycle of length e for any a, b ∈ {0, 1}. When Tc > Tes , from Property 6, one Tc of Cat map (3) over (Z2 , +, ·) satisfies has " # x 4 · NTc,e · Tc = N2Tc,e+1 · 2 · Tc, mod 2 = 0, (67) y namely 2NT ,e = N2T ,e+1. c c where Lemma 7. As for any point (x, y) in a cycle of length n of max(e , e ) if T = 1; p q c Cat map (3) over ( 2e , +, ·), Z e0 = es,g + 1 if lc = 0,T1 6= 1; " # es,g + 2 · lc − 1 if 1 ≤ lc ≤ s + 1, x e (Gn − 2) · mod 2 = 0, (66) y lc s 2 · T1 = Tc, and 2 · T1 = Tes . where n > 1. Proof. When Tc = 1 and e ≥ max(ep, eq) + 1, condition (67) should exist to satisfy Eq. (46). Setting m0 = 0 in Eq. (53), Proof. Substituting one has 1 " # " # e1 1 1 2 GT1 ≡ 1 mod 2 , 2 G2n − 1 2 Gn − 2 2 = 1 H G H e1+1 2n n n GT 6≡ 1 mod 2 , 2 1 " 1 # " # 2 Gn − 1 Gn − 2 = Gn + where Hn 0 ( 1 2 es,g if 2 GT1 ≡ 1 mod 2 ; e1 = into Eq. (62), one can obtain Eq. (66). 1 otherwise, LI et al. 13
Referring to Lemma 7, if T1 6= 1 and e ≥ es,g + 2 ≥ e1 + 2, IV. APPLICATIONOFTHECYCLESTRUCTUREOF CAT MAP any point of a cycle of length T1 should satisfy condition (67) In this section, we briefly discuss application of the obtained to meet Eq. (66). Setting m0 = 1 in Eq. (53), one has results on the cycle structure of Cat map in theoretical analysis 1 and cryptographical application. G ≡ 1 mod 2es,g , 2 2T1 The infinite number of unstable periodic orbits (UPO’s) of a 1 chaotic system constitute its skeleton [47]. In a finite-precision es,g +1 G2T 6≡ 1 mod 2 . 2 1 domain, any periodic orbit is a cycle (See Property 4). Its Referring to Lemma 4, one can further get stability is dependent on change trend of its distance with the neighboring states in the phase space. To show the relative 1 es,g +2lc positions among different cycles, we depicted real image of the G l +1 ≡ 1 mod 2 2 c T1 2 SMN shown in Fig. 1c) in Fig. 6. Note that some states may be 1 es,g +2lc+1 l +1 located in different cycles due to the finite-precision effect. The G2 c T1 6≡ 1 mod 2 2 main result obtained in [16], i.e. Eq. (6), describes the number for lc = 1 ∼ s+1. Referring to Lemma 7, if e ≥ es,g+2·lc, any of different possible Cat maps owning a specific period, which lc point of a cycle of length 2 ·T1 should satisfy condition (67) has no any relationship with the number of cycles and distance to meet Eq. (66). between a cycle and its neighboring states. So, it cannot help to identify unstable periodic orbits of the original chaotic Cat From Theorem 3, one can see that the number of cycles map at all. of various lengths in Fe can be easily deduced from that of 7 Fes when e > es. As for any cycle with length Tc ≤ Tes , 8 the threshold values of e in condition (68) are given to satisfy 6 Eq. (69). As for the cycles with length Tc > Tes , the threshold 8 values can be directly calculated with Theorem 3. As shown 5 in Theorem 4, as for every possible length of cycle, the 8 number of the cycles of the length become a fixed number 4 8 n
when e is sufficiently large. The strong regular graph patterns y 3 demonstrated in Table II and Fig. 5 are rigorously proved in 8
Theorems 3, 4. Now, we can see that the exponent value of 2 8 the distribution function of cycle lengths of Fe is fixed two e 1 when is sufficiently large. 8
Theorem 4. When 0 1 2 3 4 5 6 7 0 8 8 8 8 x 8 8 8 max(ep, eq) if Tc = 1; n Fig. 6: Portrait of Cat map (3) with N = 23 where (p, q) = es,g + 1 if lc = 0,T1 6= 1; e ≥ (68) (1, 3). e + 2 · l − 1 if 1 ≤ l ≤ s + 1; s,g c c es,g + s + 1 + lc if lc ≥ s + 2, But, knowledge of the cycle structure of Cat map can be one has used in the following aspects: • Disclosing some skeleton of Cat map in other domains. NT ,e = NT ,e+l, (69) c c As shown in Fig. 1, when p and q are both odd, there lc s where 2 ·T1 = Tc, 2 ·T1 = Tes , and l is any positive integer. is a cycle of period 3 in SMN of Cat map implemented in Galois ring Z2eˆ. This agree with the classic statement “Period three implies Chaos” given in [48]. Proof. From Property 7, one can conclude that the number of • Discarding the initial conditions corresponding to very cycles of length T in F is larger than or equal to that in c e+1 short period. This problem is vitally important in real F , i.e. N ≤ N for any e. e+1 Tc,e Tc,e+1 applications in [26], [28]. Referring to Lemma 8, as for any point (x, y) in a cycle of • Avoiding the points resulting in collision. As shown in length T = 2lc · T , c 1 Fig. 1, different points in the same cycle may evolve " x y x # " 1 # into the same point, which may result in collision for 2 2 · p · 2 − p · q · 2 2 GTc − 1 e−1 y x y · 1 mod 2 = 0 the hashing scheme proposed in [25]. 2 2 · q · 2 + p · q · 2 2 HTc • Severing as a prototype for analyzing dynamics degra- if e satisfy condition (68), meaning that NTc,e ≥ NTc,e+1. dation in chaotic maps implemented in digital computer N = N N = N l So Tc,e Tc,e+1. 2Tes ,e 2Tes ,e+l for any . From discussed in [49]. Theorem 3, when lc ≥ s + 2, e ≥ es,g + 2s + 2 + lc − s − 1 = es,g + s + 1 + lc, V. CONCLUSION
lc−s−1 This paper analyzed the structure of the 2-D generalized NT ,e = 2 · NT ,e? = NT ,e+l. c es c discrete Arnold’s Cat map by its functional graph. The explicit formulation of any iteration of the map was derived. Then, 14 IEEE TRANSACTIONS
the precise cycle distribution of the generlized discrete Cat [21] M. Farajallah, S. E. Assad, and O. Deforges, “Fast and secure chaos- map in a fixed-point arithmetic domain was derived perfectly. based cryptosystem for images,” International Journal of Bifurcation and Chaos, vol. 26, no. 2, p. art. no. 1650021, 2016. The seriously regular patterns of the phase space of Cat map [22] L. Chen and S. Wang, “Differential cryptanalysis of a medical im- implemented in digital computer were reported to dramatically age cryptosystem with multiple rounds,” Computers in Biology and different from that in the infinite-precision torus. There exists Medicine, vol. 65, pp. 69–75, 2015. [23] C. Li, D. Lin, B. Feng, J. Lu,¨ and F. Hao, “Cryptanalysis of a non-negligible number of short cycles no matter what the chaotic image encryption algorithm based on information entropy,” IEEE period of the whole Cat map is. The analysis method can be Access, vol. 6, pp. 75 834–75 842, 2018. extended to higher-dimensional Cat map and other iterative [24] C. Li, Y. Zhang, and E. Y. Xie, “When an attacker meets a cipher- image in 2018: A Year in Review,” Journal of Information Security and chaotic maps. Applications, vol. 48, p. art. no. 102361, 2019. [25] A. Kanso and M. Ghebleh, “A structure-based chaotic hashing scheme,” Nonlinear Dynamics, vol. 81, no. 1, pp. 27–40, 2015. REFERENCES [26] M. Falcioni, L. Palatella, S. Pigolotti, and A. Vulpiani, “Properties making a chaotic system a good pseudo random number generator,” [1] I. Percival and F. Vivaldi, “Arithmetical properties of strongly chaotic Physical Review E, vol. 72, no. 1, p. art. no. 016220, 2005. motions,” Physica D: Nonlinear Phenomena, vol. 25, no. 1, pp. 105– [27] T. Yarmola, “An example of a pathological random perturbation of the 130, 1987. Cat Map,” Ergodic Theory and Dynamical Systems, vol. 31, no. 6, pp. [2] D. Shi, L. L u, and G. Chen, “Totally homogeneous networks,” National 1865–1887, 2011. Science Review, p. art. no. nwz050, 2019. [28] D.-I. Curiac and C. Volosencu, “Path planning algorithm based on arnold [3] H. C. Papadopoulos and G. W. Womell, “Maximum-likelihood- cat map for surveillance uavs,” Defence Science Journal, vol. 65, no. 6, estimation of a class of chaotic signals,” IEEE Transactions on Infor- pp. 483–488, 2015. mation Theory, vol. 41, no. 1, pp. 312–317, Jan 1995. [29] C. Li, D. Lin, J. Lu,¨ and F. Hao, “Cryptanalyzing an image encryp- [4] S.-L. Chen, T. Hwang, and W.-W. Lin, “Randomness enhancement using tion algorithm based on autoblocking and electrocardiography,” IEEE digitalized modified logistic map,” IEEE Transactions on Circuits and MultiMedia, vol. 25, no. 4, pp. 46–56, 2018. Systems II-Express Briefs, vol. 57, no. 12, pp. 996–1000, Dec 2010. [30] C. Li, B. Feng, S. Li, J. Kurths, and G. Chen, “Dynamic analysis of [5] M. Garcia-Bosque, A. Perez-Resa,´ C. Sanchez-Azqueta,´ C. Aldea, digital chaotic maps via state-mapping networks,” IEEE Transactions on and S. Celma, “Chaos-based bitwise dynamical pseudorandom number Circuits and Systems I: Regular Papers, vol. 66, no. 6, pp. 2322–2335, generator on FPGA,” IEEE Transactions on Instrumentation and Mea- 2019. surement, vol. 68, no. 1, pp. 291–293, 2018. [31] B. Yang and X. Liao, “Some properties of the Logistic map over the Signal Processing [6] Z. Hua, S. Yi, Y. Zhou, C. Li, and Y. Wu, “Designing hyperchaotic cat finite field and its application,” , vol. 153, pp. 231–242, maps with any desired number of positive lyapunov exponents,” IEEE 2018. Transactions on Cybernetics, vol. 48, no. 2, pp. 463–473, Feb 2018. [32] Y. Li, “An analysis of digraphs and period properties of the logistic map on z(pn),” International Journal of Pattern Recognition and Artificial [7] T. Kohda and A. Tsuneda, “Statistics of chaotic binary sequences,” IEEE Intelligence, vol. 33, no. 3, p. art. no. 1959010, 2019. Transactions on Information Theory, vol. 43, no. 1, pp. 104–112, Jan [33] K. M. Frahm and D. L. Shepelyansky, “Small world of Ulam networks 1997. for chaotic Hamiltonian dynamics,” Physical Review E, vol. 98, no. 3, [8] K. Umeno and A.-H. Sato, “Chaotic method for generating q-gaussian p. art. no. 032205, 2018. random variables,” IEEE Transactions on Information Theory, vol. 59, [34] V. J. Garc’ia-Garrido, F. Balibrea-Iniesta, S. Wiggins, A. M. Mancho, no. 5, pp. 3199–3209, May 2013. and C. Lopesino, “Detection of phase space structures of the Cat map [9] N. Wang, C. Li, H. Bao, M. Chen, and B. Bao, “Generating multi-scroll with Lagrangian descriptors,” Regular and Chaotic Dynamics, vol. 23, Chua’s attractors via simplified piecewise-linear Chua’s diode,” IEEE no. 6, pp. 751–766, 2018. Transactions on Circuits and Systems I: Regular Papers, vol. 66, 2019. [35] D. Panario and L. Reis, “The functional graph of linear maps over finite [10] V. I. Arnol’d and A. Avez, Mathematical methods of classical mechanics. fields and applications,” Designs, Codes and Cryptography, vol. 87, no. New York: W. A. Benjamin, 1968. 2-3, pp. 437–453, 2019. [11] L. Barash and L. Shchur, “Periodic orbits of the ensemble of Sinai- [36] C. Fan and Q. Ding, “Analysing the dynamics of digital chaotic maps Arnold cat maps and pseudorandom number generation,” Physical via a new period search algorithm,” Nonlinear Dynamics, vol. 97, no. 1, Review E, vol. 73, no. 3, p. art. no. 036701, 2006. pp. 831–841, 2019. [12] O. B. Isaeva, A. Y. Jalnine, and S. P. Kuznetsov, “Arnolds cat map dy- [37] G. Chen, Y. Mao, and C. K. Chui, “A symmetric image encryption namics in a system of coupled nonautonomous van der pol oscillators,” scheme based on 3D chaotic cat maps,” Chaos Solitons & Fractals, Physical Review E, vol. 74, no. 4, p. art. no. 046207, 2006. vol. 21, no. 3, pp. 749–761, 2004. [13] L. Ermann and D. L. Shepelyansky, “The Arnold cat map, the Ulam [38] H. Kwok and W. K. Tang, “A fast image encryption system based on method and time reversal,” Physica D-Nonlinear Phenomena, vol. 241, chaotic maps with finite precision representation,” Chaos, Solitons & no. 5, pp. 514–518, 2012. Fractals, vol. 32, no. 4, pp. 1518–1529, 2007. ∗ [14] R. Okayasu, “Entropy for c -algebras with tracial rank zero,” Proceed- [39] Y. Wu, Z. Hua, and Y. Zhou, “N-dimensional discrete cat map gen- ings of the American Mathematical Society, vol. 138, no. 10, pp. 3609– eration using laplace expansions,” IEEE Transactions on Cybernetics, 3621, 2010. vol. 46, no. 11, pp. 2622–2633, Nov 2016. [15] F. Chen, K.-W. Wong, X. Liao, and T. Xiang, “Period distribution of [40] M. Sano, “Parametric dependence of the Pollicott-Ruelle resonances for e generalized discrete arnold cat map for N = p ,” IEEE Transactions sawtooth maps,” Physical Review E, vol. 66, no. 4, p. art. no. 046211, on Information Theory, vol. 58, no. 1, pp. 445–452, Jan 2012. 2002. [16] ——, “Period distribution of the generalized discrete arnold cat map for [41] F. J. Dyson and H. Falk, “Period of a discrete cat mapping,” The N = 2e,” IEEE Transactions on Information Theory, vol. 59, no. 5, pp. American Mathematical Monthly, vol. 99, no. 7, pp. 603–614, 1992. 3249–3255, May 2013. [42] J. Bao and Q. Yang, “Period of the discrete arnold cat map and general [17] P. Kurlberg and Z. Rudnick, “On the distribution of matrix elements cat map,” Nonlinear Dynamics, vol. 70, no. 2, pp. 1365–1375, 2012. for the quantum cat map,” Annals of Mathematics, vol. 161, no. 1, pp. [43] F. Chen, X. Liao, K.-W. Wong, Q. Han, and Y. Li, “Period distribution 489–507, 2005. analysis of some linear maps,” Communications in Nonlinear Science [18] M. Horvat and M. D. Esposti, “The egorov property in perturbed cat and Numerical Simulation, vol. 17, no. 10, pp. 3848–3856, Oct 2012. maps,” Journal of Physics A–Mathematical and Theoretical, vol. 40, [44] F. Chen, K.-W. Wong, X. Liao, and T. Xiang, “Period distribution of no. 32, pp. 9771–9781, 2007. generalized discrete arnold cat map,” Theoretical Computer Science, vol. [19] S. Moudgalya, T. Devakul, C. W. von Keyserlingk, and S. L. Sondhi, 552, pp. 13–25, 2014. “Operator spreading in quantum maps,” Physical Review B, vol. 99, [45] M. Hall, The Theory of Groups. The Macmillan Co., New York, 1959. no. 9, p. art. no. 094312, 2019. [46] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, [20] O. Penrose, “Entropy and irreversibility in dynamical systems,” Philo- 6th ed. Oxford OX2 6DP, UK: Oxford University Press, 2008. sophical Transactions of the Royal Society A–Mathematical Physical [47] R. L. Davidchack and Y.-C. Lai, “Efficient algorithm for detecting and Engineering Sciences, vol. 371, no. 2005, p. art. no. 20120349, unstable periodic orbits in chaotic systems,” Physical Review E, vol. 60, 2013. no. 5, pp. 6172–6175, 1999. LI et al. 15
[48] T.-Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975. [49] B. M. Boghosian, P. V. Coveney, and H. Wang, “A new pathology in the simulation of chaotic dynamical systems on digital computers,” Advanced Theory and Simulations, 2019.