A New Hyperchaotic System Generated by an External Periodic Excitation and its Image Encryption Application
Jianping Cai*,** and Jianbin He*,**,
*School of Mathematics and Statistics, Minnan Nornal University
Zhangzhou 363000, China
**Institute of Meteorological Big Data-Digital Fujian, Minnan Normal University
Zhangzhou 363000, China
By using a controller of uniformly bounded sine function, the problem of chaos anti-control for continuous linear systems is studied, and the dynamic characteristics of the controlled system are analyzed via the Lyapunov exponent spectrum and bifurcation diagram. The controlled system can be at a state of periodic motion, chaos or hyperchaos with multiple positive Lyapunov exponents when the parameters of controller belong to different intervals. Based on the hyperchaotic system, a new scheme of chaotic image encryption is proposed and it is given in the following aspects: (1) five chaotic sequences are generated from the hyperchaotic system, and the preprocessed pseudo-random sequences are used in the scrambling of the pixel positions; (2) the pixel values of image are encrypted by the combination of multiple pseudo-random sequences; (3) though the double chaotic encryption, the security of the chaotic stream cipher is analyzed by means of key sensitivity analysis, histogram analysis and information entropy analysis, etc. Finally, the experimental results show the scheme is effective and feasible in image encryption, and it can resist some attacks, such as the differential attacks, chosen-plain-text attacks, and clipping attacks.
-  G. Chen and X. Dong, “From Chaos to Order: Methodologies, Perspectives and Applications,” World Scientific Pub. Co.: Singapore, 1998.
-  T. Leung and H. Qin, “Advanced Topics in Nonlinear Control Systems,” World Scientific Pub. Co.: Singapore, 2001.
-  O. Rössler, “An Equation for Hyperchaos,” Phys. Lett. A, Vol.71, No.2-3, pp. 155-157, 1979.
-  T. Gao, Z. Chen, Z. Yuan, and G. Chen, “A Hyperchaos Generated from Chen’s System,” Int. J. Modern Phys. C, Vol.17, No.4, pp. 471-478, 2006.
-  Y. Li, W. K. Tang, and G. Chen, “Hyperchaos Evolved from the Generalized Lorenz Equation,” Int. J. Circuit Theory Appl., Vol.33, No.4, pp. 235-251, 2005.
-  T. Matsumoto, L. Chua, and K. Kobayashi, “Hyper Chaos: Laboratory Experiment and Numerical Confirmation,” IEEE Trans. Circuits Syst., Vol.33, No.11, pp. 1143-1147, 1986.
-  T. Kapitaniak, L. Chua, and G.-Q. Zhong, “Experimental Hyperchaos in Coupled Chua’s Circuits,” IEEE Trans. Circuits Syst. I, Vol.41, No.7, pp. 499-503, 1994.
-  Y. Li, W. K. Tang, and G. Chen, “Generating Hyperchaos via State Feedback Control,” Int. J. Bifurcat. Chaos, Vol.15, No.10, pp. 3367-3375, 2005.
-  R. Barboza, “Dynamics of a Hyperchaotic Lorenz System,” Int. J. Bifurcat. Chaos, Vol.17, No.12, pp. 4285-4294, 2007.
-  Y. Li, G. Chen, and W. K.-S. Tang, “Controlling a Unified Chaotic System to Hyperchaotic,” IEEE Trans. Circuits Syst. II, Vol.52, No.4, pp. 204-207, 2005.
-  J. Wang, Z. Chen, G. Chen, and Z. Yuan, “A Novel Hyperchaotic System and its Complex Dynamics,” Int. J. Bifurcat. Chaos, Vol.18, No.11, pp. 3309-3324, 2008.
-  P. C. Rech and H. A. Albuquerque, “A Hyperchaotic Chua System,” Int. J. Bifurcat. Chaos, Vol.19, No.11, pp. 3823-3828, 2009.
-  G. Hu, “Generating Hyperchaotic Attractors with Three Positive Lyapunov Exponents via State Feedback Control,” Int. J. Bifurcat. Chaos, Vol.19, No.02, pp. 651-660, 2009.
-  K. Thamilmaran, M. Lakshmanan, and A. Venkatesan, “Hyperchaos in a Modified Canonical Chua’s Circuit,” Int. J. Bifurcat. Chaos, Vol.14, No.01, pp. 221-243, 2004.
-  D. Cafagna and G. Grassi, “Decomposition Method for Studying Smooth Chua’s Equation with Application to Hyperchaotic Multiscroll Attractors,” Int. J. Bifurcat. Chaos, Vol.17, No.01, pp. 209-226, 2007.
-  M. Joshi and A. Ranjan, “New Simple Chaotic and Hyperchaotic System with an Unstable Node,” AEU-Int. J. Electronics Commun., Vol.108, pp. 1-9, 2019.
-  C. Volos, J.-O. Maaita, S. Vaidyanathan, V.-T. Pham, I. Stouboulos, and I. Kyprianidis, “A Novel Four-Dimensional Hyperchaotic Four-Wing System with a Saddle-Focus Equilibrium,” IEEE Trans. Circuits Syst. II, Vol.64, No.3, pp. 339-343, 2016.
-  S. Yu, J. Lü, X. Yu, and G. Chen, “Design and Implementation of Grid Multiwing Hyperchaotic Lorenz System Family via Switching Control and Constructing Super-Heteroclinic Loops,” IEEE Trans. Circuits Syst. I, Vol.59, No.5, pp. 1015-1028, 2012.
-  Z. Wang, S. Cang, E. O. Ochola, and Y. Sun, “A Hyperchaotic System without Equilibrium,” Nonlinear Dyn., Vol.69, No.1, pp. 531-537, 2012.
-  S. Yu and G. Chen, “Anti-Control of Continuous-Time Dynamical Systems,” Commun. Nonl. Sci. Numer. Simul., Vol.17, No.6, pp. 2617-2627, 2012.
-  C. Shen, S. Yu, J. Lü, and G. Chen, “Designing Hyperchaotic Systems with any Desired Number of Positive Lyapunov Exponents via a Simple Model,” IEEE Trans. Circuits Syst. I, Vol.61, No.8, pp. 2380-2389, 2014.
-  C. Shen, S. Yu, J. Lü, and G. Chen, “A Systematic Methodology for Constructing Hyperchaotic Systems with Multiple Positive Lyapunov Exponents and Circuit Implementation,” IEEE Trans. Circuits Syst. I, Vol.61, No.3, pp. 854-864, 2013.
-  C. Shen, S. Yu, J. Lü, and G. Chen, “Constructing Hyperchaotic Systems at Will,” Int. J. Circuit Theory Appl., Vol.43, No.12, pp. 2039-2056, 2015.
-  J. He and S. Yu, “Construction of Higher-Dimensional Hyperchaotic Systems with a Maximum Number of Positive Lyapunov Exponents under Average Eigenvalue Criteria,” J. Circuits, Syst. Comput., Vol.28, No.9, 1950151, 2019.
-  A. Daneshgar and B. Khadem, “A Self-Synchronized Chaotic Image Encryption Scheme,” Signal Processing: Image Commun., Vol.36, pp. 106-114, 2015.
-  S. Sheng, X. Zhang, and G. Lu, “Finite-Time Outer-Synchronization for Complex Networks with Markov Jump Topology via Hybrid Control and its Application to Image Encryption,” J. Franklin Institute, Vol.355, No.14, pp. 6493-6519, 2018.
-  G. Chen, Y. Mao, and C. K. Chui, “A Symmetric Image Encryption Scheme based on 3D Chaotic Cat Maps,” Chaos, Solitons & Fractals, Vol.21, No.3, pp. 749-761, 2004.
-  X. Wang and M. Zhao, “An Image Encryption Algorithm based on Hyperchaotic System and DNA Coding,” Optics & Laser Technology, Vol.143, 107316, 2021.
-  C. Pak, J. Kim, R. Pang, O. Song, H. Kim, I. Yun, and J. Kim, “A New Color Image Encryption using 2D Improved Logistic Coupling Map,” Multimedia Tools and Appl., Vol.80, pp. 25367-25387, 2021.
-  X. Wang, C. Liu, and D. Jiang, “A Novel Triple-Image Encryption and Hiding Algorithm based on Chaos, Compressive Sensing and 3D DCT,” Information Sci., Vol.574, pp. 505-527, 2021.
-  Y. Zhang, D. Xiao, W. Wen, and M. Li, “Breaking an Image Encryption Algorithm based on Hyper-Chaotic System with only One Round Diffusion Process,” Nonlinear Dyn., Vol.76, No.3, pp. 1645-1650, 2014.
-  C. Li, S. Li, and K.-T. Lo, “Breaking a Modified Substitution-Diffusion Image Cipher based on Chaotic Standard and Logistic Maps,” Commun. Nonlinear Sci. Numer. Simul., Vol.16, No.2, pp. 837-843, 2011.
-  A. Sahasrabuddhe and D. S. Laiphrakpam, “Multiple Images Encryption based on 3D Scrambling and Hyper-Chaotic System,” Information Sci., Vol.550, pp. 252-267, 2021.
-  R. Guesmi and M. B. Farah, “A New Efficient Medical Image Cipher based on Hybrid Chaotic Map and DNA Code,” Multimedia Tools and Appl., Vol.80, No.2, pp. 1925-1944, 2021.
-  S. Yu and G. Chen, “Anti-Control of Continuous-Time Dynamical Systems,” Commun. Nonlinear Sci. Numer. Simul., Vol.17, No.6, pp. 2617-2627, 2012.
This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.