Optimizing Hill Climbing Algorithm Parameters for Generation of Cryptographically Strong S-Boxes

Abstract

Nonlinear substitutions (S-boxes) are an important component of modern symmetric ciphers. They are designed to complicate the plaintext-ciphertext dependency. According to modern ideas, the S-box should be bijective, have high nonlinearity and algebraic immunity, low delta uniformity and linear redundancy. These criteria directly affect the cryptographic strength of ciphers, providing resistance to statistical, linear, algebraic, differential, and other cryptanalysis techniques. Many researchers have used various heuristic techniques to generate random S-boxes with high nonlinearity. However, the complexity of this task is still high. For example, the best-known algorithm to generate a random 8-bit bijective S-box with nonlinearity 104 requires high computational effort (i.e. more than 65 thousand intermediate estimates [search iterations]). In this article, we explore an iterative hill-climbing algorithm and optimize the heuristic search parameters. We show that the complexity of generating S-boxes can be significantly reduced. To search for a random bijective S-box with nonlinearity 104, only about 50 thousand intermediate search iterations are required. In addition, we generate Cryptographically Strong S-Boxes for which additional criteria are provided. We present estimates of the complexity of the search and estimates of the probabilities of generating substitutions with various cryptographic indicators. The extracted results demonstrate a significant improvement in our approach compared to the state of the art in terms of providing linear nonredundancy, nonlinearity (104), algebraic immunity (3) and delta uniformity (8).