A New Cryptographic Analysis of 4-bit S-Boxes

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 $$p=1/4$$p=1/4 and a minimal linear bound $$\epsilon =1/4$$∈=1/4, which imply the optimal S-boxes.

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