Substitution-boxes (S-boxes) are essential nonlinear elements in modern symmetric cryptographic systems, offering confusion and resilience against cryptanalysis. The rising need for secure communication highlights the imperative to develop resilient S-boxes with formidable cryptographic properties. This work aims to develop novel 8×8 S-boxes utilizing a hybrid approach that integrates the flower pollination algorithm (FPA) with chaotic maps. The suggested method uses a one-dimensional discrete chaotic map to generate high-quality initial S-boxes, addressing the constraints of traditional FPA techniques that may become trapped in local optima. These preliminary solutions facilitate an effective optimization procedure. The FPA metaheuristic optimizes the S-box configurations by refining a fitness function that embodies critical cryptographic requirements. The produced S-boxes are assessed using conventional metrics, such as bijectivity, rigorous avalanche criterion, nonlinearity, input/output XOR distribution, bit independence criterion, and maximal predicted linear probability. The results are juxtaposed with alternative approaches for S-box generation utilizing optimization techniques. The results demonstrate that the suggested technique generates S-boxes with robust cryptographic attributes and enhanced resilience to various forms of cryptanalytic assaults. This paper offers both theoretical and practical advances by presenting an effective hybrid optimization framework for S-box designs, enhancing the security of contemporary encryption systems.
F. Özkaynak and S. Yavuz, “Designing chaotic S-boxes based on time-delay chaotic system,” Nonlinear Dynamics, vol. 74, no. 3, pp. 551-557, 2013.
C. E. Shannon, “Communication theory of secrecy systems,” Bell Labs Technical Journal, vol. 28, no. 4, pp. 656-715, 1949.
Y. Tian and Z. Lu, “S-box: Six-dimensional compound hyperchaotic map and artificial bee colony algorithm,” Journal of Systems Engineering and Electronics, vol. 27, no. 1, pp. 232-241, 2016.
C. Carlet, “On highly nonlinear S-boxes and their inability to thwart DPA attacks,” in INDOCRYPT 2005, Springer, 2005, pp. 49-62.
S. Picek, E. Marchiori, L. Batina, and D. Jakobovic, “Combining evolutionary computation and algebraic constructions to find cryptography-relevant Boolean functions,” in International Conference on Parallel Problem Solving from Nature, Springer, 2014, pp. 822-831.
C. Carlet, “On the higher order nonlinearities of Boolean functions and S-boxes, and their generalizations,” in International Conference on Sequences and Their Applications, Springer, 2008, pp. 345-367.
X.-S. Yang, “Flower pollination algorithm for global optimization,” in International Conference on Unconventional Computing and Natural Computation, Springer, 2012, pp. 240-249.
Z. A. A. Alyasseri, A. T. Khader, M. A. Al-Betar, M. A. Awadallah, and X.-S. Yang, “Variants of the flower pollination algorithm: a review,” Nature-Inspired Algorithms and Applied Optimization, pp. 91-118, 2017.
I. Pavlyukevich, “Lévy flights, non-local search and simulated annealing,” Journal of Computational Physics, vol. 226, no. 2, pp. 1830-1844, 2007.
E. Biham and A. Shamir, “Differential cryptanalysis of DES-like cryptosystems,” in Advances in Cryptology-CRYPTO 1990, Springer, 1991, pp. 2-21.
M. Matsui, “Linear cryptanalysis method for DES cipher,” in Workshop on the Theory and Application of Cryptographic Techniques, 1993, pp. 386-397.
M. Dawson and S. E. Tavares, “An expanded set of S-box design criteria based on information theory and its relation to differential-like attacks,” in Workshop on the Theory and Application of Cryptographic Techniques, 1991, pp. 352-367.
H. A. Ahmed, M. F. Zolkipli, and M. Ahmad, “A novel efficient substitution-box design based on firefly algorithm and discrete chaotic map,” Neural Computing and Applications, pp. 1-10, 2018.
M. Ahmad, D. Bhatia, and Y. Hassan, “A novel ant colony optimization based scheme for substitution box design,” Procedia Computer Science, vol. 57, pp. 572-580, 2015.
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, vol. 88, no. 2, pp. 1059-1074, 2017.
Y. Tian and Z. Lu, “Chaotic S-Box: Intertwining Logistic Map and Bacterial Foraging Optimization,” Mathematical Problems in Engineering, 2017.
M. B. Farah, A. Farah, and T. Farah, “An image encryption scheme based on a new hybrid chaotic map and optimized substitution box,” Nonlinear Dynamics, vol. 99, no. 4, pp. 3041-3064, 2020.
H. S. Alhadawi, S. Q. Salih, and Y. D. Salman, “Chaotic particle swarm optimization based on meeting room approach for designing bijective S-boxes,” in International Conference on Emerging Technologies and Intelligent Systems, Springer, 2021, pp. 331-341.
H. S. Alhadawi, M. Ahmad, and S. Q. Salih, “A novel bijective substitution box design based on nomadic people optimizer and discrete chaotic map,” Knowledge-Based Systems, vol. 325, p. 113977, 2025.
A. Webster and S. E. Tavares, “On the design of S-boxes,” in Conference on the Theory and Application of Cryptographic Techniques, Springer, 1985, pp. 523-534.