Algorithm for Generating S-Boxes with Prescribed Differential Properties

Abstract

Cryptographic S-boxes are vectorial Boolean functions that must fulfill strict criteria to provide security for cryptographic algorithms. There are several existing methods for generating strong cryptographic S-boxes, including stochastic search algorithms. These search algorithms typically generate random candidate Boolean functions (or permutations) that are improved during the search by examining the search space in a specific way. Here, we introduce a new type of stochastic algorithm for generating cryptographic S-boxes. We do not generate and then improve the Boolean function; instead, we build the vector of values incrementally. New values are obtained by randomized search driven by restrictions on the differential spectrum of the generated S-box. In this article, we formulate two new algorithms based on this new approach and study the better one in greater detail. We prove the correctness of the proposed algorithm and evaluate its complexity. The final part contains an experimental evaluation of the method. We show that the algorithm generates S-boxes with better properties than a random search. We believe that our approach can be extended in the future by adopting more advanced stochastic search methods.