Constructing Differentially 4-uniform Permutations over GF(2 2k ) from the Inverse Function Revisited.

Abstract

Constructing S-boxes with low differential uniformity and high nonlinearity is of cardinal significance in cryptography. In the present paper, we show that numerous differentially 4-uniform permutations over F22k can be constructed by composing the inverse function and cycles over F22k . Two sufficient conditions are given, which ensure that the differential uniformity of the corresponding compositions equals 4. A lower bound on nonlinearity is also given for permutations constructed with the method in the present paper. Moreover, up to CCZ-equivalence, a new differentially 4-uniform permutation with the best known nonlinearity over F22k with k odd is constructed. For some special cycles, necessary and sufficient conditions are given such that the corresponding compositions are differentially 4-uniform.