Particle Swarm Optimization based on S-Boxes Generation

The generation of nonlinear substitutions (S-boxes) is an important task in the design of modern symmetric cryptoalgorithms. Various cryptographic properties of S-boxes (nonlinearity, balance, delta-uniformity, correlation and algebraic immunity, etc.) characterize their resistance to linear, differential, algebraic and other cryptanalysis methods. This article explores a computational particle swarm optimization (PSO) method as applied to the problem of generating nonlinear substitutions. Having a set of possible solutions (particles) and moving these particles in the search space, the PSO tries to improve the possible solution in terms of some quality indicator. We use nonlinearity, balance, delta uniformity, algebraic immunity and linear redundancy as the main indicators, and randomly generated S-boxes are used as a set of particles. This article shows several PSO modifications for generating nonlinear substitutions. At first, we reproduce the previously known PSO modification for generating S-boxes and show its low efficiency. At second, we propose our own PSO implementation and show that this method can actually generate substitutions with high cryptographic properties. The experimental results allow us to establish the influence of the size of the population of particles and the number of iterations of the outer loop on the efficiency of the heuristic generation of nonlinear substitutions. In addition, we explore the similarity of the generated substitution tables with the AES cipher S-box.