Period Distribution of Generalized Discrete Arnold Cat Map for $N=p^{e}$

In this paper, we analyze the period distribution of the generalized discrete cat map over the Galois ring where is a prime. The sequences generated by this map are modeled as 2-dimensional LFSR sequences. Employing the generation function and the Hensel lifting approaches, full knowledge of the detail period distribution is obtained analytically. Our results not only characterize the period distribution of the cat map, which gives insights to various applications, but also demonstrate some approaches to deal with the period of a polynomial in the Galois ring.

[1]  Wenbao Han,et al.  Random properties of the highest level sequences of primitive sequences over Z(2e) , 2003, IEEE Trans. Inf. Theory.

[2]  Ioannis Pitas,et al.  Region-based image watermarking , 2001, IEEE Trans. Image Process..

[3]  G. Gaspari,et al.  The Arnold cat map on prime lattices , 1994 .

[4]  L. Kocarev,et al.  Chaos and cryptography: block encryption ciphers based on chaotic maps , 2001 .

[5]  Bernold Fiedler,et al.  Periods of discretized linear Anosov maps , 1998, Ergodic Theory and Dynamical Systems.

[6]  Morgan Ward,et al.  The arithmetical theory of linear recurring series , 1933 .

[7]  YiWei Zhang,et al.  A chaos-based image encryption algorithm using alternate structure , 2007, Science in China Series F: Information Sciences.

[8]  Michael Peter Kennedy,et al.  The role of synchronization in digital communications using chaos. I . Fundamentals of digital communications , 1997 .

[9]  Rainer A. Rueppel,et al.  Products of linear recurring sequences with maximum complexity , 1987, IEEE Trans. Inf. Theory.

[10]  Michael Peter Kennedy,et al.  The role of synchronization in digital communications using chaos. I . Fundamentals of digital communications , 1997 .

[11]  D. R. Heath-Brown,et al.  An Introduction to the Theory of Numbers, Sixth Edition , 2008 .

[12]  Kwok-Wo Wong,et al.  Period Distribution of the Generalized Discrete Arnold Cat Map for $N = 2^{e}$ , 2013, IEEE Transactions on Information Theory.

[13]  Simon R. Blackburn,et al.  The linear complexity of the self-shrinking generator , 1999, IEEE Trans. Inf. Theory.

[14]  J. Keating.  Asymptotic properties of the periodic orbits of the cat maps , 1991 .

[15]  Michael Peter Kennedy,et al.  Chaos shift keying : modulation and demodulation of a chaotic carrier using self-sychronizing chua"s circuits , 1993 .

[16]  Cunsheng Ding,et al.  The Stability Theory of Stream Ciphers , 1991, Lecture Notes in Computer Science.

[17]  Der-Chyuan Lou,et al.  A steganographic scheme for secure communications based on the chaos and euler Theorem , 2004, IEEE Transactions on Multimedia.

[18]  Ezra Brown,et al.  Cycles of directed graphs defined by matrix multiplication (mod n) , 2001, Discret. Math..

[19]  Peter Seibt.  A period formula for torus automorphisms , 2003 .

[20]  Michael Peter Kennedy,et al.  The role of synchronization in digital communications using chaos. II. Chaotic modulation and chaotic synchronization , 1998 .

[21]  Ljupco Kocarev,et al.  Public-key encryption with chaos. , 2004, Chaos.

[22]  Ranjan Bose,et al.  Novel public key encryption technique based on multiple chaotic systems. , 2005, Physical review letters.

[23]  Edwin L. Key,et al.  An analysis of the structure and complexity of nonlinear binary sequence generators , 1976, IEEE Trans. Inf. Theory.

[24]  Z. Guan,et al.  Chaos-based image encryption algorithm ✩ , 2005 .

[25]  Tore Herlestam,et al.  On Functions of Linear Shift Register Sequences , 1985, EUROCRYPT.

[26]  I. Percival,et al.  Arithmetical properties of strongly chaotic motions , 1987 .

[27]  R. A. Rueppel.  Analysis and Design of Stream Ciphers , 2012 .

[28]  J. Fridrich.  Symmetric Ciphers Based on Two-Dimensional Chaotic Maps , 1998 .

[29]  Zhe-Xian X. Wan,et al.  Quaternary Codes , 1997 .

[30]  Wen Feng Qi,et al.  Further Result of Compressing Maps on Primitive Sequences Modulo Odd Prime Powers , 2007, IEEE Transactions on Information Theory.

[31]  R. Tennant.  Algebra , 1941, Nature.

[32]  Anne Canteaut.  Fast correlation attacks against stream ciphers and related open problems , 2005, IEEE Information Theory Workshop on Theory and Practice in Information-Theoretic Security, 2005..

[33]  Michael Peter Kennedy,et al.  Communications using chaos/spl Gt/MINUS. III. Performance bounds for correlation receivers , 2000 .

[34]  Zhi-Hong Guan,et al.  A novel digital watermark algorithm based on chaotic maps , 2007 .

[35]  M Huang.  MAXIMAL PERIOD POLYNOMIALS OVER Z/(p~d) , 1992 .

[36]  Hongxia Wang,et al.  Public Watermarking Based on Chaotic Map , 2004 .

[37]  F. Dyson,et al.  Period of a Discrete Cat Mapping , 1992 .

[38]  Patrick Solé,et al.  Quaternary Convolutional Codes From Linear Block Codes Over Galois Rings , 2007, IEEE Transactions on Information Theory.

[39]  Henry Leung,et al.  Ergodic chaotic parameter modulation with application to digital image watermarking , 2005, IEEE Transactions on Image Processing.

[40]  Kwok-Wo Wong,et al.  Period Distribution of Generalized Discrete Arnold Cat Map for N=pe , 2012, IEEE Trans. Inf. Theory.

[41]  Zongduo Dai,et al.  Binary sequences derived from ML-sequences over rings I: Periods and minimal polynomials , 1992, Journal of Cryptology.

[42]  Vera Pless,et al.  Cyclic codes and quadratic residue codes over Z4 , 1996, IEEE Trans. Inf. Theory.

[43]  Riccardo Rovatti,et al.  Chaotic complex spreading sequences for asynchronous DS-CDMA. I. System modeling and results , 1997 .

[44]  V. I. Arnolʹd,et al.  Ergodic problems of classical mechanics , 1968 .

[45]  Z. Wan.  Lectures on Finite Fields and Galois Rings , 2003 .

[46]  R. Rovatti,et al.  Chaotic complex spreading sequences for asynchronous DS-CDMA. Part II. Some theoretical performance bounds , 1998 .

[47]  Guang Gong,et al.  Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar , 2005 .

[48]  Matthew J. B. Robshaw,et al.  New Stream Cipher Designs: The eSTREAM Finalists , 2008 .

[49]  W. Schwarz,et al.  Chaos communications-principles, schemes, and system analysis , 2002, Proc. IEEE.

[50]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[51]  Solomon W. Golomb,et al.  Shift Register Sequences , 1981 .

[52]  Michael Baake,et al.  Periodic orbits of linear endomorphisms on the 2-torus and its lattices , 2008, 0808.3489.

[53]  C. Chui,et al.  A symmetric image encryption scheme based on 3D chaotic cat maps , 2004 .

[54]  Ljupco Kocarev,et al.  Public-key encryption based on Chebyshev maps , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..