New Families of Differentially 4-Uniform Permutations over ${\mathbb F}_{2^{2k}}$

Abstract

Differentially 4-uniform permutations over ${\mathbb F}_{2^{2k}}$, especially those with high nonlinearity and high algebraic degree, are cryptographically significant mappings as they are good choices for the substitution boxes (S-boxes) in many symmetric ciphers. For instance, the currently endorsed Advanced Encryption Standard (AES) uses the inverse function, which is a differentially 4-uniform permutation. However, up to now, there are only five known infinite families of such mappings which attain the known maximal nonlinearity. Most of these five families have small algebraic degrees and only one family can be defined over ${\mathbb F}_{2^{2k}}$ for any positive integer k. In this paper, we apply the powerful switching method on the five known families to construct differentially 4-uniform permutations. New infinite families of such permutations are discovered from the inverse function, and some sporadic examples are found from the others by using a computer. All newly found infinite families can be defined over fields ${\mathbb F}_{2^{2k}}$ for any k and their algebraic degrees are 2k − 1. Furthermore, we obtain a lower bound for the nonlinearity of one infinite family.