Implicit quadratic property of differentially 4-uniform permutations

Abstract

Substitution box (S-box) is an important component of block ciphers for providing nonlinearity. It is often constructed from differentially 4-uniform permutation. In this paper, we examine all (to the best of our knowledge) the differentially 4-uniform permutations that are known in the literature and determine whether they are implicitly quadratic. We found that all of them are implicitly quadratic, making them vulnerable to algebraic attack [10, 12–14]. This leads to an open question of whether there exists a differentially 4-uniform permutation over F2n that is not implicitly quadratic. We provide a partial answer to this question by solving it for the special cases of n = 11 and n = 13.