Discrete Chaotic Systems with One-Line Equilibria and Their Application to Image Encryption

This paper introduces nine four-dimensional discrete chaotic systems with one-line equilibria (DCSLE), consisting of some simple sine functions. Based on the generalized chaos synchronization (GCS) theorem, a DCSLE is used to construct an eight-dimensional DCSLE GCS system. The new DCSLE GCS system is verified by numerical simulation and then used to design a chaotic pseudorandom number generator (CPRNG). The randomness of ten 100-key streams generated by the CPRNG, two GCS-based CPRNGs, the RC4 PRNG and the ZUC PRNG are tested by the SP800-22/FIPS 140-2 tests. The test results confirm that the randomness performances of the three CPRNGs are promising, for there are no significant correlations between a keystream and any perturbed keystream generated by such CPRNG. Also, the key space of the CPRNG is larger than 21170. Finally, the CPRNG is used with an avalanche-effect encryption scheme to encrypt an RGB image, demonstrating that the CPRNG is able to generate the avalanche effects which are similar to those generated by ideal CPRNGs.

[1]  Recai Kiliç,et al.  Experimental study on impulsive synchronization between two modified Chua's circuits , 2006 .

[2]  J. Sprott,et al.  Some simple chaotic flows. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  A. Kanso,et al.  A novel image encryption algorithm based on a 3D chaotic map , 2012 .

[4]  Yi Shen,et al.  Compound synchronization of four memristor chaotic oscillator systems and secure communication. , 2013, Chaos.

[5]  K. Wong,et al.  A secure communication scheme based on the phase synchronization of chaotic systems. , 2003, Chaos.

[6]  J. M. Sausedo-Solorio,et al.  Synchronization of map-based neurons with memory and synaptic delay , 2014 .

[7]  Wei Pan,et al.  Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers , 2014, Commun. Nonlinear Sci. Numer. Simul..

[8]  Lequan Min,et al.  A generalized synchronization theorem for discrete-time chaos system with application in data encryption scheme , 2007, 2007 International Conference on Communications, Circuits and Systems.

[9]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[10]  Zheng-Ming Ge,et al.  Rapid Communication Chaos synchronization of double Duffing systems with parameters excited by a chaotic signal , 2008 .

[11]  A. E. Matouk,et al.  Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit , 2011 .

[12]  Cheng-Hsiung Yang,et al.  Chaos generalized synchronization of an inertial tachometer with new Mathieu-Van der Pol systems as functional system by GYC partial region stability theory , 2012 .

[13]  Lequan Min,et al.  A Generalized Stability Theorem for Discrete-Time Nonautonomous Chaos System with Applications , 2015 .

[14]  Yong Wang,et al.  A new chaos-based fast image encryption algorithm , 2011, Appl. Soft Comput..

[15]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[16]  M. Senator,et al.  Synchronization of two coupled escapement-driven pendulum clocks , 2006 .

[17]  Marcelo Messias,et al.  Hopf bifurcation from Lines of Equilibria without Parameters in Memristor oscillators , 2010, Int. J. Bifurc. Chaos.

[18]  Lequan Min,et al.  Design of Non-autonomous Chaotic Generalized Synchronization Based Pseudorandom Number Generator with Application in Avalanche Image Encryption , 2014, 2014 IEEE 17th International Conference on Computational Science and Engineering.

[19]  Faqiang Wang,et al.  A new criterion for chaos and hyperchaos synchronization using linear feedback control , 2006 .

[20]  Wu Xiang-Jun,et al.  A new chaotic communication scheme based on adaptive synchronization. , 2006, Chaos.

[21]  Lequan Min,et al.  Study on the Statistical Test for String Pseudorandom Number Generators , 2013, BICS.

[22]  Lequan Min,et al.  A novel stream encryption scheme with avalanche effect , 2013 .

[23]  Lequan Min,et al.  A Stream Encryption Scheme with Both Key and Plaintext Avalanche Effects for Designing Chaos-Based Pseudorandom Number Generator with Application to Image Encryption , 2016, Int. J. Bifurc. Chaos.

[24]  Eduardo Serrano,et al.  Synchronization effects using a piecewise linear map-based spiking-bursting neuron model , 2006, Neurocomputing.

[25]  Xiangjun Wu,et al.  A new color image cryptosystem via hyperchaos synchronization , 2014, Commun. Nonlinear Sci. Numer. Simul..

[26]  Paul Woafo,et al.  Chaotic synchronization with experimental application to secure communications , 2009 .

[27]  Lee Sun-Jin.  From Chaos to Order , 2011 .

[28]  Julien Clinton Sprott,et al.  Simple chaotic flows with a line equilibrium , 2013 .

[29]  Lequan Min,et al.  A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption. , 2015, Chaos.

[30]  Parlitz,et al.  Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. , 1996, Physical review letters.

[31]  G Alvarez,et al.  Breaking a secure communication scheme based on the phase synchronization of chaotic systems. , 2003, Chaos.

[32]  Yicong Zhou,et al.  Cascade Chaotic System With Applications , 2015, IEEE Transactions on Cybernetics.

[33]  Wolfgang Kinzel,et al.  Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers , 2006, nlin/0604068.

[34]  Bernold Fiedler,et al.  Generic Hopf bifurcation from Lines of Equilibria without parameters III: Binary oscillations , 2000, Int. J. Bifurc. Chaos.

[35]  Julien Clinton Sprott,et al.  Elementary quadratic chaotic flows with no equilibria , 2013 .

[36]  J. Alexander,et al.  Generic Hopf Bifurcation from Lines of Equilibria without Parameters: I. Theory☆ , 2000 .

[37]  U. Dieckmann,et al.  POPULATION GROWTH IN SPACE AND TIME: SPATIAL LOGISTIC EQUATIONS , 2003 .

[38]  Bernold Fiedler,et al.  Generic Hopf Bifurcation From Lines of Equilibria Without Parameters: II. Systems of Viscous Hyperbolic Balance Laws , 2000, SIAM J. Math. Anal..

[39]  Fabricio A. Breve,et al.  Chaotic phase synchronization and desynchronization in an oscillator network for object selection , 2009, Neural Networks.

[40]  H. Feistel.  Cryptography and Computer Privacy , 1973 .

[41]  Sundarapandian Vaidyanathan,et al.  Analysis, synchronization and circuit design of a novel butterfly attractor , 2014 .

[42]  Leon O. Chua,et al.  Chua's circuit 10 years later , 1994, Int. J. Circuit Theory Appl..

[43]  Elaine B. Barker,et al.  A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications , 2000 .

[44]  Jinde Cao,et al.  Adaptive synchronization of a switching system and its applications to secure communications. , 2008, Chaos.

[45]  K. A. Shore,et al.  Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization. , 2009 .

[46]  J. Sprott.  Chaos and time-series analysis , 2001 .

[47]  Yicong Zhou,et al.  $n$ -Dimensional Discrete Cat Map Generation Using Laplace Expansions , 2016, IEEE Transactions on Cybernetics.

[48]  Nikolay V. Kuznetsov,et al.  Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems , 2011 .