Period of the discrete Arnold cat map and general cat map

The paper first studies the period of the discrete Arnold cat map. When the modulo is composite, the formulae are developed to calculate the minimal period. When the modulo is prime, the formulae calculating the period are given and an algorithm is proposed in order to determine the minimal period. Then the paper explores the relationship between the period of the discrete general cat map and its modulo for different parameters a and b. Some period formulae are given and some properties about the period are obtained. In addition, the paper also expands the period formulae of the corresponding a-Fibonacci sequence taken the modulo for more new parameters.

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

[2]  Gabriel Peterson,et al.  Arnold's Cat Map , 1997 .

[3]  Wei Pang,et al.  Dynamics of the general Lorenz family , 2012 .

[4]  Xiaoling Huang,et al.  Image encryption algorithm using chaotic Chebyshev generator , 2011, Nonlinear Dynamics.

[5]  Qigui Yang,et al.  Dynamical analysis of the generalized Sprott C system with only two stable equilibria , 2012 .

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

[7]  Josef Scharinger.  Fast encryption of image data using chaotic Kolmogorov flows , 1998, J. Electronic Imaging.

[8]  Ángel Plaza,et al.  k-Fibonacci sequences modulo m , 2009 .

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

[10]  M. Yaghoobi,et al.  A new image encryption method: parallel sub-image encryption with hyper chaos , 2011, Nonlinear Dynamics.

[11]  Wei Zhang,et al.  A chaos-based symmetric image encryption scheme using a bit-level permutation , 2011, Inf. Sci..

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

[13]  Kwok-Wo Wong,et al.  An efficient chaotic image encryption algorithm based on a generalized Arnold map , 2012, Nonlinear Dynamics.

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

[15]  D. D. Wall.  Fibonacci Series Modulo m , 1960 .

[16]  Xing-yuan Wang,et al.  A chaotic image encryption algorithm based on perceptron model , 2010 .

[17]  Guanrong Chen,et al.  Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems , 2005 .