Counteracting Dynamical Degradation of Digital Chaotic Chebyshev Map via Perturbation

The chaotic map has complex dynamics under ideal conditions however it suffers from the problem of performance degradation in the case of finite computing precision. In order to prevent the dynamics degradation, in this paper the continuous Chen chaotic system is used to perturb both the inputs and parameters of Chebyshev map to minimize the chaotic degradation phenomenon under finite precision. Experimental evaluations and corresponding performance analysis demonstrate that the Chebyshev chaotic map has a good randomness and complex dynamic performance by using the proposed perturbation method, and some attributes of the proposed system are stronger than the original system (e.g. chaos attractor and approximate entropy). Finally, the corresponding pseudorandom number generator (PRNG) is constructed by this method and then its randomness is evaluated via NIST SP800-22 and TestU01 test suites, respectively. Statistical test results show that the proposed PRNG has high reliability of randomness, thus it can be used for cryptography and other potential applications.

[1]  Shujun Li,et al.  Cryptanalysis of a new chaotic cryptosystem based on ergodicity , 2008, 0806.3183.

[2]  Jinyu Kuang,et al.  PERIODICITY OF CHAOTIC TRAJECTORIES IN REALIZATIONS OF FINITE COMPUTER PRECISIONS AND ITS IMPLICATION IN CHAOS COMMUNICATIONS , 2003, nlin/0309005.

[3]  W. Kinsner.  Characterizing chaos through Lyapunov metrics , 2006, IEEE Trans. Syst. Man Cybern. Syst..

[4]  S M Pincus,et al.  Approximate entropy as a measure of system complexity. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[5]  X. Mou,et al.  On the security of a chaotic encryption scheme: problems with computerized chaos in finite computing precision , 2003 .

[6]  Yuling Luo,et al.  A perturbation method to the tent map based on Lyapunov exponent and its application , 2015 .

[7]  Xing-yuan Wang,et al.  A new pseudo-random number generator based on CML and chaotic iteration , 2012 .

[8]  Pierre L'Ecuyer,et al.  TestU01: A C library for empirical testing of random number generators , 2006, TOMS.

[9]  Anastasios Bezerianos,et al.  Secure Communication with Chaotic Systems of Difference Equations , 1997, IEEE Trans. Computers.

[10]  Jacques M. Bahi,et al.  Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems , 2015, IEEE Transactions on Circuits and Systems I: Regular Papers.

[11]  Di Xiao,et al.  Cryptanalysis of S-box-only chaotic image ciphers against chosen plaintext attack , 2013, Nonlinear Dynamics.

[12]  Minghui Du,et al.  A symmetrical image encryption scheme in wavelet and time domain , 2015, Commun. Nonlinear Sci. Numer. Simul..

[13]  Xiao-Jun Tong,et al.  Design of an image encryption scheme based on a multiple chaotic map , 2013, Commun. Nonlinear Sci. Numer. Simul..

[14]  Kwok-Wo Wong,et al.  A Fast Image Encryption Scheme based on Chaotic Standard Map , 2006, ArXiv.

[15]  Naixue Xiong,et al.  Analysis and Design of Digital Chaotic Systems With Desirable Performance via Feedback Control , 2015, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[16]  Lingfeng Liu,et al.  An analogue-digital mixed method for solving the dynamical degradation of digital chaotic systems , 2014, IMA J. Math. Control. Inf..

[17]  Xiaole Fang,et al.  Theoretical Design and Circuit Implementation of Integer Domain Chaotic Systems , 2014, Int. J. Bifurc. Chaos.

[18]  Hongjun Liu,et al.  Color image encryption based on one-time keys and robust chaotic maps , 2010, Comput. Math. Appl..

[19]  Fei Xiang,et al.  Analysis on Stability of Binary Chaotic Pseudorandom Sequence , 2008, IEEE Communications Letters.

[20]  A. Kanso,et al.  Logistic chaotic maps for binary numbers generations , 2009 .

[21]  Junxiu Liu,et al.  Design and Implementation of Image Chaotic Communication via FPGA Embedded Ethernet Transmission , 2009, 2009 International Workshop on Chaos-Fractals Theories and Applications.

[22]  Sang Tao,et al.  Perturbance-based algorithm to expand cycle length of chaotic key stream , 1998 .

[23]  Yongduan Song,et al.  Fuzzy variable structure control for uncertain systems with disturbance , 2012 .

[24]  In-Kwon Lee,et al.  Chaotic image encryption using pseudo-random masks and pixel mapping , 2016, Signal Process..

[25]  Wolfgang A. Halang,et al.  Cryptanalysis of an image encryption scheme based on a compound chaotic sequence , 2007, Image Vis. Comput..

[26]  Greg M. Bernstein,et al.  Secure random number generation using chaotic circuits , 1990 .

[27]  Clare D. McGillem,et al.  A chaotic direct-sequence spread-spectrum communication system , 1994, IEEE Trans. Commun..

[28]  Magdi S. Mahmoud,et al.  Robust H∞ reliable control for uncertain switched neutral systems with distributed delays , 2015, IMA J. Math. Control. Inf..

[29]  Lingfeng Liu,et al.  A universal method for improving the dynamical degradation of a digital chaotic system , 2015 .

[30]  Dahua Song,et al.  A New Property of Logistic Map with Scalable Precision , 2012, 2012 Fifth International Workshop on Chaos-fractals Theories and Applications.

[31]  Chengqing Li,et al.  Deciphering an Image Cipher Based on Mixed Transformed Logistic Maps , 2014, Int. J. Bifurc. Chaos.

[32]  Tsin-Yuan Chang,et al.  A chaos-based pseudo random number generator using timing-based reseeding method , 2006, 2006 IEEE International Symposium on Circuits and Systems.

[33]  M. Perc.  Visualizing the attraction of strange attractors , 2005 .

[34]  Tao Xie,et al.  Breaking a novel image encryption scheme based on improved hyperchaotic sequences , 2012, Nonlinear Dynamics.

[35]  Yicong Zhou,et al.  Image encryption using 2D Logistic-adjusted-Sine map , 2016, Inf. Sci..

[36]  Daniel D. Wheeler,et al.  Supercomputer Investigations of a Chaotic Encryption Algorithm , 1991, Cryptologia.

[37]  Yicong Zhou,et al.  Dynamic Parameter-Control Chaotic System , 2016, IEEE Transactions on Cybernetics.

[38]  L. Kocarev.  Chaos-based cryptography: a brief overview , 2001 .

[39]  Guanrong Chen,et al.  Design and ARM-Embedded Implementation of a Chaotic Map-Based Real-Time Secure Video Communication System , 2015, IEEE Transactions on Circuits and Systems for Video Technology.

[40]  Ziqi Zhu,et al.  A method of improving the properties of digital chaotic system , 2008 .

[41]  J. Cernák.  Digital generators of chaos , 1996 .

[42]  Lingfeng Liu,et al.  Counteracting the dynamical degradation of digital chaos via hybrid control , 2014, Commun. Nonlinear Sci. Numer. Simul..

[43]  Wolfgang A. Halang,et al.  Cryptanalysis of an image encryption scheme based on a new total shuffling algorithm , 2007, ArXiv.

[44]  Massimo Alioto,et al.  A Class of Maximum-Period Nonlinear Congruential Generators Derived From the Rényi Chaotic Map , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.