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Period Distribution of the Generalized Discrete Arnold Cat Map for Fei Chen, Kwok-Wo Wong, Senior Member, IEEE, Xiaofeng Liao, and Tao Xiang

Abstract—The Arnold cat map is employed in various applications where chaos is utilized, especially chaos-based cryptography and watermarking. In this paper, we study the problem of period distribution of the generalized discrete Arnold cat map over the . Full knowledge of the period distribution is obGalois ring tained analytically by adopting the Hensel lift approach. Our results have impact on both chaos theory and its applications as they not only provide design strategy in applications where special periods are required, but also help to identify unstable periodic orbits of the original chaotic cat map. The method in our paper also shows some ideas how to handle problems over the Galois ring . Index Terms—Galois ring LFSR, period distribution.

, generalized cat map, Hensel lift,

I. INTRODUCTION N recent years, there is a large amount of work utilizing chaos in various algorithms and systems for communication [1]–[3], cryptography [4]–[12], and watermarking [13]–[15]. The Arnold cat map [16] is often employed as it possesses nice ergodic and mixing properties. This map is an area-preserving chaotic map having the form

I

(1) . In practical applications such as where watermarking [13], image encryption [6], and public-key cryptosystem [10], the original cat map is generalized and discretized to the following form: (2) where

.

Manuscript received June 15, 2012; revised December 05, 2012; accepted December 11, 2012. Date of publication December 21, 2012; date of current version April 17, 2013. This work was supported in part by a grant from Research Grants Council of the Hong Kong Special Administrative Region, China, through Project CityU 123009, in part by the National Natural Science Foundation of China under Grants 60973114 and 61103211, and in part by the Post-doctoral Science Foundation of China under Grant 201104319. F. Chen is with the Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong (e-mail: [email protected]. hk). K.-W. Wong is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). X. Liao is with the School of Electronic and Information Engineering, Southwest University, Chongqing 400716, China, and also with the College of Computer Science, Chongqing University, Chongqing 400030, China (e-mail: [email protected]). T. Xiang is with the College of Computer Science, Chongqing University, Chongqing 400044, China (e-mail: [email protected]). Communicated by T. Helleseth, Associate Editor for Sequences. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2012.2235907

In applications such as watermarking or image encryption, usually denotes the initial position of an image pixel, while represents the position of the pixel after iterations of the cat map. In other applications such as private-key and public-key cryptosystems, partial or the whole part of and the number of iterations play the role of the secret key. It is obvious that the cat map must have a period after discretization, which may become a major design consideration in certain applications. For example, the cryptographic algorithms [10], [12] and steganographic algorithms [13] are vulnerable to attacks if the period of the underlying chaotic map is short. The detailed period distribution is thus of crucial importance in the design of various chaos-based systems. This knowledge also contributes to chaos theory in understanding the unstable rational periodic orbits of (1). The reason is that if a rational point is a periodic point of (1), then is also a periodic point of (2) and vice versa. The sequences generated by the cat map (2) can be modeled as LFSR sequences and, thus, can be analyzed using the generating function approach. The period distribution problem can be systematically investigated in three cases, i.e., forms a Galois field, a Galois ring, and a general commutative ring, which correspond to the cases as is a prime, is a power of a prime, and is a composite. The case when is a prime is easy to be analyzed since the Galois field has a perfect structure. The case when is a composite is also easy to be analyzed if the results for the former two cases are obtained, which is due to the Chinese Remainder Theorem. For the case that forms a Galois ring, i.e., where is a prime, things become complicated. In this case, we have obtained analytical results for by combining the generating function method and the Hensel lift method [17]. However, the approach failed for the case when . Here, we analyze the period distribution for the generalized discrete cat map (2) when , i.e., to obtain exact statistics on the period of the cat map when and traverse all elements in . Some previous work on this topic has been reported [18]–[20]. However, only partial distribution results for the special case were obtained. For general cat maps with arbitrary and , the exact period distribution remains an open problem. For the case of , it turns out surprisingly that the distribution problem can be solved completely only with some elementary algebra. The detailed mathematical analysis is simple and easy to follow. Our approach employs the Hensel lift method [21], [22]. These results help in the period distribution analysis of the cat maps for arbitrary . This paper is organized as follows. In Section II, the approach for investigating the period distribution of generalized discrete cat map is introduced, followed by the basic idea employed in our analysis. The detailed analysis work is carried out

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in Sections III and IV. The complete results are presented at the end of Section IV, together with some implications of our work. Conclusions are made in Section V. II. PRELIMINARIES AND BASIC IDEA This section introduces some concepts and notations in understanding our analysis. For more about number theory and abstract algebra, see [22]–[24]. When , the cat map has the form (3) where

. Let

. We

assume throughout this paper because the cases when and are trivial. First, the period distribution problem is stated. Let be an initial point of cat map (3) and be the point after processed by iterations of the cat map from . If there exists an integer such that for all initial points in , is called the period of the cat map (3). This period must exist since . The period distribution analysis is then reduced to finding all the periods and the number of distinct maps possessing a specific period when and traverse all possible values in . Second, we introduce the recurrence equation theory as well as some concepts and notations which are useful in understanding our analysis. forms a Galois ring where addition and multiplication are all modular operations. Let and denote the greatest common divisor and the least common multiple of and , respectively. means that is a divisor of . Suppose that is a polynomial in and , the period of , denoted as , is defined as the smallest integer such that where all the arithmetic operations are over . A typical recurrence equation over is (4) . Let be a sequence generated where by (4). The generating function of is defined as . Let . A simple calculation results in and so (5) . The polynomial is where called the characteristic polynomial of the recurrence (4). If and are coprime over , is called the minimal polynomial of (4). Suppose that the period of is , i.e., for . It leads to which results in another form of : (6) where

.

; thus, Equations (5) and (6) lead to . If and are coprime, and the following well-known proposition must hold [25], [26]. and Proposition 1[25], [26]: Let be the period of its generating function be . If and are coprime, . Third, the basic idea for analyzing the period distribution of the cat map (3) is the use of the above recurrence equation theory to model this problem. Let and be the sequences generated by (3). Let and be their generating functions, respectively. Then, it holds that and which result in (7) and (8) where (9) , and . For a particular initial point , if or is coprime with , the period of this point is from Proposition 1. Observing that if is odd, the point is special as which is coprime with and the period of this point is . If is odd, the point is also special as which is coprime with . The period of this point is also . However, such a point cannot be found when and are both even. It is easy to verify that the period of the cat map is the least common multiple of the periods for all initial points. Thus, if and are not both even, the period must be . However, if and are both even, the situation is different and it only holds that the period of the cat map must be a divisor of . Therefore, it is natural to have individual analysis for these two cases. The analysis in each case is composed of two steps: the period analysis step finds out all possible periods, while the counting step counts how many distinct cat maps possessing a specific period. It helps if there is an impression on how the period distribution looks like. Fig. 1 shows such an example for . It can be observed that the periods distribute in very sparsely and the maximal period is 16 which equals . The number of cat maps having this period is not small. In the following sections, the period distribution rules will be worked out theoretically. III. PERIOD DISTRIBUTION WHEN ARE NOT BOTH EVEN

AND

If and are not both even, the period of cat map (3) equals to . For prime , is a finite field and it is easy to analyze the period of a polynomial in the finite field. However, is a finite ring (exactly a Galois ring), whose structure is more complicated. There are zero divisors in which make the analysis quite different. An elementary

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where

. Squaring both sides of (12), we have , where . This process repeats and we obtain (13)

. From (13), it is where as in easy to verify that which implies . This is the spirit of Hensel’s lift. Whether can reach the maximal value desince . pends on If , and the maximal value is . It is obvious that (14) Fig. 1. Period distribution for

.. .

.

approach, the Hensel lift approach [21], [22], is adopted here to analyze the period distribution in . Hensel lift is a common method in number theory and Galois ring theory. Its basic idea is informally described as follows. Given a problem in , we first study it in which is a Galois field having a good structure, then extend the results to . In [27]–[29], Hensel lift approach was adopted to study the maximal period of a linear recurrence. It is also used to study nonlinear codes over [21]. We find that the Hensel lift approach gives satisfactory results in our period distribution analysis. The outline of our approach is described as follows. The main idea is to analyze the period of over . First, its period is studied in . Then, the result is extended to and the number of cat maps possessing each possible period is counted. Let be the expression (9) and be the polynomial of when it is considered as a polynomial in . can have two forms, i.e., and , which are analyzed individually. A. The Case That If in , in the counting step. Let

in which will help (10)

where (11) Thus, in . It is worth noting that condition (11) is quite critical as it will play an important role in counting how many cat maps possessing a certain period. Let the period of in be . It is easy to observe that since divides in . Actually, in . This is the period of in . Now we lift this result into . Considering and as polynomials in leads to . In , it holds that (12)

since Thus, if

vanishes under the modulo 2 operation. , we must have and . It is obvious that must not be 0 modulo . Now, it is natural to perform the analysis in two different cases. 1) Case 1: : Summarizing the above discussions, the following proposition holds. Proposition 2: If , and there are cat maps having such a period. Proof: Period analysis. From the above discussion, it holds that in . While , it is true that , in , and is the smallest integer that such relationship holds. Therefore, . Counting. From (10), . Moreover, satisfies condition (11) and . It is trivial to verify that the sufficient and necessary condition for , where , is that is odd. This implies . Based on this condition, we can count the number of cat maps with this period. If , there are choices for . Then, 4 must divide and the number of possible is . As a result, there are a total of cat maps in this case. If , it must hold that since and are not both even which is the underlying condition. Then, it follows that . Similar to the above case, there are cat maps satisfying this condition. Combining the above results, there are cat maps having period . 2) Case 2: : In this case, must be even and the period of must be less than . The exact period depends on and it is not a hard problem if the Hensel lift process is repeated as in (12) and (13). The following are the detailed results for and , respectively. , Proposition 3: If maps possessing this period.

and there are

cat

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Proof: Period analysis. From (12), gives in . Squaring both sides gets . This also holds in and so . Counting. From (10), we have , where . If is odd with possible choices, is uniquely determined by . As a result, there are cat maps under this condition. If is even, it must hold that where is odd. The number of choices for is . Once is fixed, there are only two possible values for : or . There are cat maps in this case. Combining the above results, there are cat maps with period 4. Proposition 4: Let is an integer and traverses the set cat maps having period Proof: Period analysis. From (12), Squaring both sides leads to

, where is odd by condition (11). and there are for each .

TABLE I PERIOD DISTRIBUTION FOR THE CASE

IN

where (16) Thus, . Like (11), condition (16) is also critical in the counting step. In , it holds that since due to the factorization . Now the analysis can proceed with a Hensel lift process. Considering and as polynomials in , it gives . In , it holds that (17)

.

Let

where (18)

is an integer, and (19)

where gives

means that

. Continuing in this way and . Similar to (14), it holds that since is even and is odd. This and so

. Counting. From (10), , where is an integer and . Notice that . Using the same argument as for Proposition 3, there are cat maps of period for each , given a fixed . Moreover, being odd means that the number of choices for is . As a result, there are cat maps having period for each . Now the period distribution for the case in is clear. Combining Propositions 2–4, we summarize our partial results as follows.

is odd by condition (16). This leads to . It turns out that the period distribution strongly depends on and our results are stated as follows. , goes over the set and there are cat maps of period for each . If , there are cat maps with period 3. Proof: Period analysis. From (17), . Let . Squaring both sides gives Proposition 6: If

(20) where leads to

Theorem 5: Let be the period of the cat map (3). If in , the possible periods and the corresponding number of cat maps having such periods are given by Table I. B. The Case That

in

In this case, counting step. Let

which also helps in the

.. .

Thus, integer such that (15)

. Continuing in this fashion and . Since , it is easy to verify that

and in

is the smallest . Thus

CHEN et al.: PERIOD DISTRIBUTION OF THE GENERALIZED DISCRETE ARNOLD CAT MAP FOR

Counting. From (15), and must be odd. Notice the condition that , and is odd. If , must be 1 and there are choices for . Once is chosen, is determined by . So there are cat maps of period 3. If , must belong to . There are choices for since it is odd. Given a fixed , using the same argument as above, there are cat maps of period . Thus, the total number of cat maps corresponding to this situation is . Notice that Proposition 6 requires . If , it is easy to observe which implies that the period is smaller. However, it holds, in this situation, that . Since is odd, must be even. There are two different cases: and . In the latter situation, let where and is odd by condition (19). Now we have

where

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TABLE II PERIOD DISTRIBUTION FOR THE CASE

IN

cat maps having period for each . The case . In this case, . Notice that must be odd. There are choices for . Once is determined, is also uniquely determined. Thus, there are cat maps with period 6. Now the period distribution for the case in is analyzed. Combining Propositions 6 and 7, we summarize our partial results as follows. Theorem 8: Let

be the period of the cat map (3). If in , the possible periods and the corresponding number of cat maps possessing such periods are listed in Table II. IV. PERIOD DISTRIBUTION WHEN

AND

ARE

BOTH EVEN

When

This turns out to be the same situation as Propositions 3, 4, and the corresponding proofs apply here. Therefore, we have the following results. Proposition 7: Suppose . For , let . Then, goes over the set and there are cat maps having period for each . If , and there are cat maps in this situation. Proof: Period analysis. The case . Suppose . In this case, we have

and sides gives and

and are both even, it holds that and . If , it is trivial that the period of the cat map is 1. In the following analysis, we assume at least one of and is nonzero. Now the period of the cat map may not be equal to and so another way to deal with the period distribution analysis is required. First, the period of the cat map is reconsidered. Rewrite the cat map (3) as . Suppose its period is , then . Since is the period, holds and it follows that in for all possible initial points in , where is the identity matrix and is the zero vector. It seems that should hold. A more detailed analysis confirms this. Proposition 9: The period of the cat map (3) is the smallest integer satisfying in . Proof: (i) Suppose is the period of the cat map. It holds that

. Repeated squaring both . Notice that . This means that and so .

The case

. In this case, we have

Thus, Counting. The case

.

and

. From (15), and must be odd. For a fixed and , there are cat maps of period . Moreover, being odd means that the number of choices for is . As a result, there are

(21) for all possible initial points in . Let , respectively. Applying them to (21) gets . Thus, . (ii) Suppose . It is valid that for all possible initial points in . Thus, is the period of the cat map. The proof completes by combining (i) and (ii).

in and

Proposition 9 suggests another way to compute the period, i.e., to find the smallest integer such that in . In general, this is somewhat difficult and the carry problem in the arithmetic operations needs to be solved. But for the case with and both even, this approach can give satisfactory results by combining with the Hensel lift method. We show this as follows.

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Let the 2-adic expansion of and be and , respectively, where . Since and are both even, then , and the 2-adic expansion of is

TABLE III PERIOD DISTRIBUTION FOR

Rewriting this expression leads to (22) and

where

is the th term in the 2-adic

expansion of . Computing the difficult now once (22) is obtained.

th power of

is not

Proposition 10: For the cat map (3) with given by (22), its period traverses the set and there are cat maps corresponding to period . Proof: Period analysis. Suppose is the first term in (22) whose elements are not all zeros. In this situation, it must hold that since and the th term in its 2-adic expansion must be 0. From , squaring both sides gets , where

There are only three possible values for , and

TABLE IV EXPERIMENTAL RESULTS FOR

, i.e.,

,

. A direct computation verifies that

in any case. Continuing in this manner gives in and is the smallest integer such that this equality holds. According to Proposition 9, the period is also . When varies from 1 to , traverses the set . Counting. Let where and be the first term in (22) whose elements are not all zeros. If , the number of possible is while the number of choices for is . In this case, there are cat maps. If , must be 1. The number of choices for and are the same, i.e., . There are cat maps satisfying this condition. Thus, there are cat maps of period where . This means that there are cat maps possessing period where . Now we have discussed all possible cases for analyzing the period distribution of the cat map. Combining Theorems 5, 8, and Proposition 10, we summarize our results in the following theorem. Notice that we also need to take into account the trivial case and . where and be the period Theorem 11: Let of cat map over . The possible periods and the number of cat maps corresponding to those periods are listed in Table III.

TABLE V EXPERIMENTAL RESULTS FOR

Remark 12: There are cat maps in , which implies that summation of the last column of Table III should be . This is verified here. It is easy to find out that and . Now adding all entries in the last column of Table III gives . This shows the correctness of our analysis in another way. Example 13: Two examples are given here to show the consistency of the experimental and the theoretical results. A computer program has been written to exhaust all possible cat maps over and . The period of each cat map is computed by brute force and the results are listed in Tables IV and V, respectively. For , it is easy to observe that the maximal period is 16 which is equal to . The number of cat maps having this period is 64 which is equal to . There are also cat maps of period 6. The same also applies to . The experimental results are consistent with the theoretical ones. Table III contains our main results on the period distribution of the cat map over . Much information can be derived from it. The maximal period is and there are cat maps possessing this period. This amount is exactly a quarter of the total number of possible cat maps and is not small. The practical impact of our results is described as follows. From the cryptanalytic point of view, some existing steganographic and cryptographic systems such as in [10], [12], and [13] are vulnerable to attacks if the period of the underlying chaotic map is not sufficiently large. From the design point of view, our results lead to the choice of proper cat maps when a specific period, no matter large or small, is required.

CHEN et al.: PERIOD DISTRIBUTION OF THE GENERALIZED DISCRETE ARNOLD CAT MAP FOR

V. CONCLUSION The period distribution of the cat map over is analyzed in detail using the Hensel lift method. Full knowledge of this distribution is obtained, which helps in the design and analysis of watermarking and cryptographic algorithms using cat maps. Our next step aims to work out the corresponding period distribution for general composite ’s. REFERENCES [1] G. Kolumban, M. Kennedy, and L. Chua, “The role of synchronization in digital communications using chaos. I. Fundamentals of digital communications,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., vol. 44, no. 10, pp. 927–936, Oct. 1997. [2] G. Kolumban, M. Kennedy, and L. Chua, “The role of synchronization in digital communications using chaos. II. Chaotic modulation and chaotic synchronization,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., vol. 45, no. 11, pp. 1129–1140, Nov. 1998. [3] G. Kolumban and M. Kennedy, “The role of synchronization in digital communications using chaos-Part III: Performance bounds for correlation receivers,” IEEE Trans. Circuits Syst., vol. 47, no. 12, pp. 1673–1683, Dec. 2000. [4] J. Fridrich, “Symmetric ciphers based on two-dimensional chaotic maps,” Int. J. Bifurcat. Chaos, vol. 8, pp. 1259–1284, 1998. [5] G. Jakimoski and L. Kocarev, “Chaos and cryptography: Block encryption ciphers based on chaotic maps,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., vol. 48, no. 2, pp. 163–169, Feb. 2001. [6] G. Chen, Y. Mao, and C. Chui, “A symmetric image encryption scheme based on 3D chaotic cat maps,” Chaos, Solitons Fractals, vol. 21, no. 3, pp. 749–761, 2004. [7] A. N. Pisarchik, N. J. Flores-Carmona, and M. Carpio-Valadez, “Encryption and decryption of images with chaotic map lattices,” Chaos: Interdisciplinary J. Nonlinear Sci., vol. 16, no. 3, pp. 033118-1–033118-6, 2006. [8] T. Xiang, K. Wong, and X. Liao, “Selective image encryption using a spatiotemporal chaotic system,” Chaos: Interdisciplinary J. Nonlinear Sci., vol. 17, pp. 023115-1–023115-12, 2007. [9] R. Hasimoto-Beltran, “High-performance multimedia encryption system based on chaos,” Chaos: Interdisciplinary J. Nonlinear Sci., vol. 18, no. 2, pp. 023110-1–023110-8, 2008. [10] L. Kocarev, M. Sterjev, A. Fekete, and G. Vattay, “Public-key encryption with chaos,” Chaos: Interdisciplinary J. Nonlinear Sci., vol. 14, pp. 1078–1082, 2004. [11] X. Wang, X. Gong, M. Zhan, and C. H. Lai, “Public-key encryption based on generalized synchronization of coupled map lattices,” Chaos: Interdisciplinary J. Nonlinear Sci., vol. 15, no. 2, pp. 023109-1–023109-8, 2005. [12] R. Bose, “Novel public key encryption technique based on multiple chaotic systems,” Phys. Rev. Lett., vol. 95, no. 9, pp. 98702–98705, 2005. [13] D. Lou and C. Sung, “A steganographic scheme for secure communications based on the chaos and Euler theorem,” IEEE Trans. Multimedia, vol. 6, no. 3, pp. 501–509, Jun. 2004. [14] S. Chen and H. Leung, “Ergodic chaotic parameter modulation with application to digital image watermarking,” IEEE Trans. Image Process., vol. 14, no. 10, pp. 1590–1602, Oct. 2005. [15] X. Wu and Z. Guan, “A novel digital watermark algorithm based on chaotic maps,” Phys. Lett. A, vol. 365, no. 5–6, pp. 403–406, 2007. [16] V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics. New York: Benjamin, 1968. [17] F. Chen, K. Wong, X. Liao, and T. Xiang, “Period distribution of gener,” IEEE Trans. Inf. Theory, alized discrete Arnold cat map for vol. 58, no. 1, pp. 445–452, Jan. 2012.

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[18] I. Percival and F. Vivaldi, “Arithmetical properties of strongly chaotic motions,” Physica D: Nonlinear Phenomena, vol. 25, no. 1–3, pp. 105–130, 1987. [19] J. Keating, “Asymptotic properties of the periodic orbits of the cat maps,” Nonlinearity, vol. 4, pp. 277–307, 1991. [20] F. Dyson and H. Falk, “Period of a discrete cat mapping,” Amer. Math. Monthly, vol. 99, no. 7, pp. 603–614, 1992. [21] Z. Wan, Quaternary Codes. Singapore: World Scientific, 1997. [22] Z. Wan, Lectures on Finite Fields and Galois Rings. Singapore: World Scientific, 2003. [23] G. Hardy, E. Wright, D. Heath-Brown, and J. Silverman, An Introduction to the Theory of Numbers. Oxford, U.K.: Clarendon, 1960. [24] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications. Cambridge, U.K.: Cambridge Univ. Press, 1994. [25] S. Golomb and L. Welch, Shift Register Sequences. Laguna Hills, CA: Aegean Park, 1982. [26] S. Golomb and G. Gong, Signal Design for Good Correlation: for Wireless Communication, Cryptography, and Radar. Cambridge, U.K.: Cambridge Univ. Press, 2005. [27] M. Ward, “The arithmetical theory of linear recurring series,” Trans. Amer. Math. Soc., vol. 35, no. 3, pp. 600–628, 1933. [28] Z. Dai and M. Huang, “A criterion for primitiveness of polynomial over ,” Chin. Sci. Bull., vol. 36, pp. 892–895, 1991. ,” Sci. China, Series [29] M. Huang, “Maximal period polynomials over A, vol. 35, pp. 271–275, 1992. Fei Chen received the B.E. and M.E. degrees in computer science and engineering from Chongqing University, China, in 2008 and 2011, respectively. He is now a Ph.D. student in The Chinese University of Hong Kong. His research interests include algorithms, cryptography and network security. His current personal webpage is https://sites.google.com/site/chenfeiorange/. Kwok-Wo Wong (SM’03) received the B.Sc. degree in electronic engineering from The Chinese University of Hong Kong and the Ph.D. degree from City University of Hong Kong, where he is currently an associate professor in Department of Electronic Engineering. His current research interests include chaos, cryptography, and source coding. He has published more than 100 papers in mathematics, physics, and engineering journals in the fields of nonlinear dynamics, cryptography, neural networks, and optics. He is an associate editor of International Journal of Bifurcation and Chaos (IJBC), an editor of The HKIE Transactions, and Mathematical Problems in Engineering. Xiaofeng Liao received the BS and MS degrees in mathematics from Sichuan University, Chengdu, China, in 1986 and 1992, respectively, and the Ph.D. degree in circuits and systems from the University of Electronic Science and Technology of China in 1997. From 1999 to 2012, he is a professor at Chongqing University. At present, he is a professor at Southwest University and the Dean of School of Electronic and Information Engineering. He is also a Yangtze River Scholar of the Ministry of Education of China. From November 1997 to April 1998, he was a research associate at the Chinese University of Hong Kong. From October 1999 to October 2000, he was a research associate at the City University of Hong Kong. From March 2001 to June 2001 and March 2002 to June 2002, he was a senior research associate at the City University of Hong Kong. From March 2006 to April 2007, he was a research fellow at the City University of Hong Kong. He is also awarded of Prize for Natural Science from the Ministry of Education and Chongqing respectively. Professor Liao holds 4 patents, and published 3 books and over 250 international journal and conference papers. His current research interests include neural networks, nonlinear dynamical systems, bifurcation and chaos, and cryptography. Tao Xiang received the B.S., M.S. and Ph.D. degrees in computer science from Chongqing University, China, in 2003, 2005, and 2008, respectively. He is currently an Associate Professor of Chongqing University. His research interests include multimedia security, chaotic cryptography, and particle swarm optimization. He has published more than 30 papers on international journals and conferences. He also served as a referee for numerous international journals.