A new cryptographic analysis of 4-bit S-boxes

Abstract

An exhaustive search of all 16! bijective 4-bit S-boxes has been conducted by Markku-Juhani et al. (SAC 2011). In this paper, we present an improved exhaustive search over all permutation-xor equivalence classes. We put forward some optimizing strategies and make some improvements on the basis of their work. For our program, it only takes about one-sixth of the time of the experiment by Markku-Juhani et al. to get the same results. Furthermore, we classify all those permutation-xor equivalence classes in terms of a new classification criterion, which has been come up with by Wentao Zhang et al. (FSE 2015). For some special cases, we calculate the distributions of permutation-xor equivalence classes with respect to their differential bound and linear bound. It turns out that only in three special cases, there exist S-boxes having a minimal differential bound p = 1/4 and a minimal linear bound ϵ = 1/4, which imply the optimal S-boxes.