Classification of $$6\times 6$$ S-boxes Obtained by Concatenation of RSSBs

Abstract

We give an efficient exhaustive search algorithm to enumerate \(6\times 6\) bijective S-boxes with the best known nonlinearity 24 in a class of S-boxes that are symmetric under the permutation \(\tau (x) = (x_0, x_2, x_3, x_4,\) \(x_5, x_1)\), where \(x = (x_0,\) \(x_1, \ldots , x_5) \in \mathbb {F}_{2}^6\). Since any S-box \(S: \mathbb {F}_{2}^6\rightarrow \mathbb {F}_{2}^6\) in this class has the property that \(S(\tau (x))=\tau (S(x))\) for all x, it can be considered as a construction obtained by the concatenation of \(5\times 5\) rotation-symmetric S-boxes (RSSBs). The size of the search space, i.e., the number of S-boxes belonging to the class, is \(2^{61.28}\). By performing our algorithm, we find that there exist \(2^{37.56}\) S-boxes with nonlinearity 24 and among them the number of differentially 4-uniform ones is \(2^{33.99}\), which indicates that the concatenation method provides a rich class in terms of high nonlinearity and low differential uniformity. Moreover, we classify those S-boxes achieving the best possible trade-off between nonlinearity and differential uniformity within the class with respect to absolute indicator, algebraic degree, and transparency order.