Generating dynamical S-boxes using 1D Chebyshev chaotic maps

Document Type: Original Article


Department of Computer Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.



This paper considers the construction of chaotic dynamic substitution boxes (S-boxes) using chaotic Chebyshev polynomials of the first kind. The proposed algorithm provides dynamic S-boxes with acceptable security performance compared to twenty-one recent schemes. This algorithm is used to generate 80 random 8 × 8 S-boxes and analyzed their security performance. Their average performance shows acceptable security. The security of the generated S-boxes is measured against several mandatory security requirements for S-box designs including bijective property, strict avalanche criterion (SAC), linear approximation probability (LAP), differential approximation probability, bit independence criterion, correlation immunity, algebraic immunity, auto-correlation, and propagation criterion. Moreover, the set of majority logic criterion measures is used to measure the quality and robustness of the generated S-boxes in image encryption. Because obtaining a chaotic sequence with one dimensional Chebyshev polynomials of the first kind is very simpler and efficient than the hyper-chaotic mappings, the proposed algorithm is of lower computational costs compared with recent chaotic S-box generation algorithms.


[1] National Institute of Standards and Technology. Data Encryption Standard (DES). FIPS, 46-3, 1999. [ bib ]
[2] J. Daemen and V. Rijmen. he design of : AES-the advanced encryption standard. Springer Science Business Media, 2013. [ bib ]
[3] C. E. Shannon. Communication theory of secrecy systems. Bell System Technical Journal, 28(4):656--715, 1949. [ bib | DOI ]
[4] S. Shaukat Jamal, M. Khan Usman, and T. Shah. A watermarking technique with chaotic fractional S-box transformation. Wireless Personal Communications, 90(4):2033–2049, 2016. [ bib | DOI ]
[5] B. R. Gangadari and S. R. Ahamed. Programmable cellular automata-based low-power architecture to S-box: An application to WBAN. Circuits, Systems, and Signal Processing, 37(3):1116–1133, 2018. [ bib | DOI ]
[6] T. WCusick and P. Stanica. Cryptographic Boolean functions and applications. Academic Press, 2017. [ bib ]
[7] F. Özkaynak and A. B. Özer. A method for designing strong s-boxes based on chaotic Lorenz system. Physics Letters A, 374(36):3733--3738, 2010. [ bib | DOI ]
[8] I. Hussain, T. Shah, M. A. Gondal, W. A. Khan, and H. Mahmood. A group theoretic approach to construct cryptographically strong substitution boxes. Neural Computing and Applications, 23(1):97–104, 2012. [ bib | DOI ]
[9] S. Farwa, T. Shah, and L. Idrees. A highly nonlinear S-box based on a fractional linear transformation. SpringerPlus, 5(1):1658, 2016. [ bib | DOI ]
[10] S. Farwa, N. Muhammad, T. Shah, and S. Ahmad. A novel image encryption based on algebraic S-box and Arnold transform. 3D Research, 8(3):26, 2017. [ bib | DOI ]
[11] M. Khan and Z. Asghar. A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and S 8 permutation. Neural Computing and Applications, 29(4):993–999, 2018. [ bib | DOI ]
[12] J. A. Aboytes-González, J. S. Murguía, M. Mejía-Carlos, H. González-Aguilar, and M. T. Ramírez-Torres. Design of a strong S-box based on a matrix approach. Nonlinear Dynamics, 94(3):2003--2012, 2018. [ bib | DOI ]
[13] T. Ye and L. Zhimao. Chaotic S-box: six-dimensional fractional Lorenz--Duffing chaotic system and O-shaped path scrambling. Nonlinear Dynamics, 94(3):2115–2126, 2018. [ bib | DOI ]
[14] Ü. Çavusoğlu, S. Kaçar, A. Zengin, and I. Pehlivan. A novel hybrid encryption algorithm based on chaos and S-AES algorithm. Nonlinear Dynamics, 92(4):1745--1759, 2018. [ bib | DOI ]
[15] A. Shakiba. Security analysis for chaotic maps-based mutual authentication and key agreement using smart cards for wireless networks. Journal of Information and Optimization Sciences, 40(3):725--750, 2019. [ bib | DOI ]
[16] A. Shakiba. A randomized CPA-secure asymmetric-key chaotic color image encryption scheme based on the Chebyshev mappings and one-time pad. Journal of King Saud University-Computer and Information Sciences, pages 725--750, 2019. [ bib | DOI ]
[17] A. Shakiba. A novel randomized one-dimensional chaotic Chebyshev mapping for chosen plaintext attack secure image encryption with a novel chaotic breadth first traversal. Multimedia Tools and Applications, 78(24):34773--34799, 2019. [ bib | DOI ]
[18] A. Shakiba, M. R. Hooshmandasl, and M. A. Meybodi. Cryptanalysis of multiplicative coupled cryptosystems based on the chebyshev polynomials. International Journal of Bifurcation and Chaos, 26(07):1650112, 2016. [ bib | DOI ]
[19] G. Tang, X. Liao, and Y. Chen. A novel method for designing S-boxes based on chaotic maps. Chaos, Solitons & Fractals, 23(2):413--419, 2005. [ bib | DOI ]
[20] A. Belazi and A. A. A. El-Latif. A simple yet efficient S-box method based on chaotic sine map. Optik, 130:1438--1444, 2017. [ bib | DOI ]
[21] C. Pak and L. Huang. A new color image encryption using combination of the 1D chaotic map. Signal Processing, 138:129--137, 2017. [ bib | DOI ]
[22] G. Chen, Y. Chen, and X. Liao. An extended method for obtaining S-boxes based on three-dimensional chaotic baker maps. Chaos, solitons & fractals, 31(3):571--579, 2007. [ bib | DOI ]
[23] M. Khan, T. Shah, H. Mahmood, M. A. Gondal, and I. Hussain. A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dynamics, 70(3):2303–2311, 2012. [ bib | DOI ]
[24] F. Özkaynak, V. Çelik, and A. B. Özer. A new S-box construction method based on the fractional-order chaotic chen system. Signal, Image and Video Processing, 11(4):659–664, 2017. [ bib | DOI ]
[25] I. Hussain, T. Shah, and M. A. Gondal. A novel approach for designing substitution-boxes based on nonlinear chaotic algorithm. Nonlinear Dynamics, 70(3):1791–1794, 2012. [ bib | DOI ]
[26] M. Khan, T. Shah, and M. A. Gondal. An efficient technique for the construction of substitution box with chaotic partial differential equation. Nonlinear Dynamics, 73(3):1795–1801, 2013. [ bib | DOI ]
[27] A. Anees and Z. Ahmed. A technique for designing substitution box based on Vanderpol oscillator. Wireless Personal Communications, 82(3):1497–1503, 2015. [ bib | DOI ]
[28] Ü. Çavusoğlu, A. Zengin, I. Pehlivan, and S. Kaçar. A novel approach for strong S-box generation algorithm design based on chaotic scaled ZhongTang system. Nonlinear Dynamics, 87(2):1081–1094, 2017. [ bib | DOI ]
[29] T. Ritter. Substitution cipher with pseudo-random shuffling: The dynamic substitution combiner. Cryptologia, 14(4):289--303, 1990. [ bib | DOI ]
[30] T. Ritter. Transposition cipher with pseudo-random shuffling: The dynamic transposition combiner. Cryptologia, 15(1):1--17, 1991. [ bib | DOI ]
[31] D. Guo, L. Cheng, and L. Cheng. A new symmetric probabilistic encryption scheme based on chaotic attractors of neural networks. Applied Intelligence, 10(1):71–84, 1991. [ bib | DOI ]
[32] J. Urias, E. Ugalde, and G. Salazar. A cryptosystem based on cellular automata. Chaos: An Interdisciplinary Journal of Nonlinear Science, 8(4):819--822, 1998. [ bib | DOI ]
[33] S. Li, X. Zheng, X. Mou, and Y. Cai. Chaotic encryption scheme for real-time digital video. In Real-Time Imaging VI, pages 149--160. International Society for Optics and Photonics, 2002. [ bib | DOI ]
[34] G. Tang and X. Liao. A method for designing dynamical S-boxes based on discretized chaotic map. Chaos, solitons & fractals, 23(5):1901--1909, 2005. [ bib | DOI ]
[35] J. C. Mason and D. C. Handscomb. Chebyshev polynomials. CRC press, 2002. [ bib ]
[36] K. Briggs. An improved method for estimating Liapunov exponents of chaotic time series. Physics Letters A, 151(1-2):27--32, 1990. [ bib | DOI ]
[37] J. Wu, X. Liao, and B. Yang. Image encryption using 2D Hénon-sine map and DNA approach. Signal Processing, 153:11--23, 2018. [ bib | DOI ]
[38] C. Li, T. Xie, Q. Liu, and G. Cheng. Cryptanalyzing image encryption using chaotic Logistic map. Nonlinear Dynamics, 78(2):1545–1551, 2014. [ bib | DOI ]
[39] R. A. Elmanfaloty and E. Abou-Bakr. Random property enhancement of a 1D chaotic PRNG with finite precision implementation. Chaos, Solitons & Fractals, 118(2):134--144, 2019. [ bib | DOI ]
[40] I. Hussain and T. Shah. Literature survey on nonlinear components and chaotic nonlinear components of block ciphers. Nonlinear Dynamics, 74(4):869–904, 2013. [ bib | DOI ]
[41] C. Adams and S. Tavares. The structured design of cryptographically good S-boxes. Journal of Cryptology, 3(1):27–41, 1990. [ bib | DOI ]
[42] W. Millan, A. Clark, and E. Dawson. An effective genetic algorithm for finding highly nonlinear Boolean functions. In International Conference on Information and Communications Security, pages 149--158. Springer, Berlin, Heidelberg, 1997. [ bib | DOI ]
[43] I. Hussain, T. Shah, M. A. Gondal, and H. Mahmood. Generalized majority logic criterion to analyze the statistical strength of S-boxes. Zeitschrift für Naturforschung A, 67(5):282--288, 2012. [ bib | DOI ]
[44] A. Webster and S. E. Tavares. On the design of S-boxes. In Conference on the theory and application of cryptographic techniques, pages 523--534. Springer, Berlin, Heidelberg, 1985. [ bib | DOI ]
[45] E. Biham and A. Shamir. Differential cryptanalysis of DES-like cryptosystems. Journal of Cryptology, 4(1):3–72, 1991. [ bib | DOI ]
[46] F. Firdousi, S. I. Batool, and M. Amin. A novel construction scheme for nonlinear component based on quantum map. International Journal of Theoretical Physics, 58(11):3871--3898, 2019. [ bib | DOI ]
[47] A. Rukhin, J. Soto, J. Nechvatal, M. Smid, and E. Barker. A statistical test suite for random and pseudorandom number generators for cryptographic applications. Technical report, Booz-allen and hamilton inc mclean va, 2001. [ bib ]
[48] T. Farah, R. Rhouma, and S. Belghith. A novel method for designing S-box based on chaotic map and teaching–learning-based optimization. Nonlinear Dynamics, 88(2):1059--1074, 2017. [ bib | DOI ]
[49] M. Khan, T. Shah, and S. I. Batool. Construction of S-box based on chaotic boolean functions and its application in image encryption. Neural Computing and Applications, 27(3):677--685, 2016. [ bib | DOI ]
[50] M. Khan. A novel image encryption scheme based on multiple chaotic S-boxes. Nonlinear Dynamics, 82(1-2):527--533, 2015. [ bib | DOI ]
[51] M. Khan and T. Shah. A construction of novel chaos base nonlinear component of block cipher. Nonlinear Dynamics, 76(1):377--382, 2014. [ bib | DOI ]
[52] G. Chen. A novel heuristic method for obtaining S-boxes. Chaos, Solitons & Fractals, 36(4):1028--1036, 2008. [ bib | DOI ]