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.