Abstract
This paper discusses the implementation sensitivity of chaotic systems added to their widely discussed sensitivities to initial conditions and parameter variation. This sensitivity can cause mismatches in some applications, which require an exact duplication of the system, e.g., chaos-based cryptography, synchronization and communication. Specifically, different implementation cases of three discretized jerk-based chaotic systems and a discrete-time logistic map are presented corresponding to different orders of additions and multiplications. The cases exhibit roughly similar attractor shapes, bifurcation behavior and Lyapunov exponents. However, mismatches between the time series corresponding to these cases in software double-precision, single-precision floating-point and hardware fixed-point implementations are reported. The number of time units after which the mismatch starts to become noticeable, and the effects of the discretization step and precision are discussed. Experimental results on Artix-7 XC7A100T FPGA and oscilloscope validate the presence of mismatch reported through simulations. The wrong decryption effect of this mismatch is demonstrated for a software image encryption application, where one case is used for encryption and the other(s) for decryption. Pseudo-Random Number Generation and image encryption application using the mismatch signal as a chaotic generator are proposed and show good results using several well-established performance metrics.
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A.T. Azar, C. Volos, N.A. Gerodimos, G.S. Tombras, V.T. Pham, A.G. Radwan, S. Vaidyanathan, A. Ouannas, J.M. Munoz-Pacheco, A novel chaotic system without equilibrium: dynamics, synchronization, and circuit realization. Complexity (2017). https://doi.org/10.1155/2017/7871467
M.L. Barakat, A.S. Mansingka, A.G. Radwan, K.N. Salama, Generalized hardware post-processing technique for chaos-based pseudorandom number generators. ETRI J. 35(3), 448–458 (2013)
M.L. Barakat, A.S. Mansingka, A.G. Radwan, K.N. Salama, Hardware stream cipher with controllable chaos generator for colour image encryption. IET Image Proc. 8(1), 33–43 (2014)
I. Dassios, D. Baleanu, Optimal solutions for singular linear systems of caputo fractional differential equations. Math. Methods Appl. Sci. (2018). https://doi.org/10.1002/mma.5410
I. Dassios, K. Fountoulakis, J. Gondzio, A preconditioner for a primal-dual newton conjugate gradient method for compressed sensing problems. SIAM J. Sci. Comput. 37(6), A2783–A2812 (2015)
I.K. Dassios, Analytic loss minimization: theoretical framework of a second order optimization method. Symmetry 11(2), 136 (2019)
A. Elwakil, K. Salama, M. Kennedy, A system for chaos generation and its implementation in monolithic form. in International Symposium on Circuits and Systems (ISCAS), vol 5 (IEEE, 2000), pp. 217–220
D. Goldberg, What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. (CSUR) 23(1), 5–48 (1991)
V.N. Govorukhin, Calculation lyapunov exponents for ODE. MATLAB Central File Exchange, file ID 4628 (2004)
A.S. Mansingka, A.G. Radwan, M.A. Zidan, K. Salama, Analysis of bus width and delay on a fully digital signum nonlinearity chaotic oscillator. in 54th International Midwest Symposium on Circuits and Systems (MWSCAS) (IEEE, 2011), pp. 1–4
D. Monniaux, The pitfalls of verifying floating-point computations. ACM Trans. Program. Lang. Syst. (TOPLAS) 30(3), 12 (2008)
L.G. Nardo, E.G. Nepomuceno, J. Arias-Garcia, D.N. Butusov, Image encryption using finite-precision error. Chaos Solitons Fractals 123, 69–78 (2019)
E.G. Nepomuceno, S.A. Martins, B.C. Silva, G.F. Amaral, M. Perc, Detecting unreliable computer simulations of recursive functions with interval extensions. Appl. Math. Comput. 329, 408–419 (2018)
G. Peng, F. Min, Multistability analysis, circuit implementations and application in image encryption of a novel memristive chaotic circuit. Nonlinear Dyn. 90(3), 1607–1625 (2017)
A.G. Radwan, S.H. AbdElHaleem, S.K. Abd-El-Hafiz, Symmetric encryption algorithms using chaotic and non-chaotic generators: a review. J. Adv. Res. 7, 193–208 (2015)
A.G. Radwan, A.M. Soliman, A.L. El-Sedeek, An inductorless CMOS realization of Chua’s circuit. Chaos Solitons Fractals 18(1), 149–158 (2003)
R. Rakkiyappan, R. Sivasamy, X. Li, Synchronization of identical and nonidentical memristor-based chaotic systems via active backstepping control technique. Circ. Syst. Signal Process. 34(3), 763–778 (2015)
A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, A statistical test suite for random and pseudorandom number generators for cryptographic applications (Tech. rep, DTIC Document, 2001)
W.S. Sayed, H.A. Fahmy, A.A. Rezk, A.G. Radwan, Generalized smooth transition map between tent and logistic maps. Int. J. Bifurc. Chaos 27(01), 1730004 (2017)
W.S. Sayed, M.M. Henein, S.K. Abd-El-Hafiz, A.G. Radwan, Generalized dynamic switched synchronization between combinations of fractional-order chaotic systems. Complexity 2017, (2017). https://doi.org/10.1155/2017/9189120
W.S. Sayed, A.G. Radwan, H.A. Fahmy, Design of positive, negative, and alternating sign generalized logistic maps. Discrete Dyn. Nat. Soc. 2015, (2015). https://doi.org/10.1155/2015/586783
W.S. Sayed, A.G. Radwan, A.A. Rezk, H.A. Fahmy, Finite precision logistic map between computational efficiency and accuracy with encryption applications. Complexity 2017, (2017). https://doi.org/10.1155/2017/8692046
W.S. Sayed, M.F. Tolba, A.G. Radwan, S.K. Abd-El-Hafiz, FPGA realization of a speech encryption system based on a generalized modified chaotic transition map and bit permutation. Multimed. Tools Appl. 78(12), 16097–16127 (2019)
A. Senouci, A. Boukabou, K. Busawon, A. Bouridane, A. Ouslimani, Robust chaotic communication based on indirect coupling synchronization. Circ. Syst. Signal Process. 34(2), 393–418 (2015)
J.C. Sprott, A new class of chaotic circuit. Phys. Lett. A 266(1), 19–23 (2000)
S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, Boulder, 2014)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)
F. Wu, P. Zhou, A. Alsaedi, T. Hayat, J. Ma, Synchronization dependence on initial setting of chaotic systems without equilibria. Chaos Solitons Fractals 110, 124–132 (2018)
Fq Wu, J. Ma, Gd Ren, Synchronization stability between initial-dependent oscillators with periodical and chaotic oscillation. J. Zhejiang Univ. Sci. A 19(12), 889–903 (2018)
X. Ye, X. Wang, S. Gao, J. Mou, Z. Wang, F. Yang, A new chaotic circuit with multiple memristors and its application in image encryption. Nonlinear Dyn. 99, 1–18 (2019)
M.A. Zidan, A.G. Radwan, K.N. Salama, The effect of numerical techniques on differential equation based chaotic generators. in International Conference on Microelectronics (ICM) (IEEE, 2011), pp. 1–4
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The authors would like to thank Eng. Merna Habib, Research Assistant at Nanoelectronics Integrated Systems Center, Nile University.
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Sayed, W.S., Radwan, A.G., Fahmy, H.A.H. et al. Software and Hardware Implementation Sensitivity of Chaotic Systems and Impact on Encryption Applications. Circuits Syst Signal Process 39, 5638–5655 (2020). https://doi.org/10.1007/s00034-020-01424-8
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DOI: https://doi.org/10.1007/s00034-020-01424-8