Results on rotation-symmetric S-boxes

We give an efficient exhaustive search strategy to enumerate 6 × 6 bijective rotation-symmetric S-boxes (RSSBs) having nonlinearity 24, which is found to be the maximum nonlinearity within the class of 6 × 6 bijective RSSBs. It is shown that there are 3072 RSSBs achieving the cryptographic properties of the inverse function over GF(26), i.e., nonlinearity 24, differential uniformity 4, and algebraic degree 5, such that among them there are only four which are not affine-equivalent. Among these four RSSBs, we find a non-affine transformation under which the cryptographic properties of the inverse function are invariant. Then, we define the generalized classes of k-RSSBs as the polynomials of GF(2n) with coefficients in GF(2k), where k divides n. Moreover, motivated by the fact that RSSBs are symmetric under a special permutation, we classify all possible permutations up to the linear equivalence of S-boxes that are symmetric under them.