1An Equivalent Condition on the Switching Construction of Differentially 4-uniform Permutations on F22k from the Inverse Function

Differentially 4-uniform permutations on F22k with high nonlinearity are often chosen as Substitution boxes in block ciphers. Recently, Qu et al. used the powerful switching method to construct such permutations from the inverse function [9], [10]. More precisely, they studied the functions of the form G(x) = 1x +f( 1 x), where f is a Boolean function. They proved that if f is a preferred Boolean function (PBF), then G is a permutation polynomial over F2n whose differential uniformity is at most 4. However, as pointed out in [9], f is a PBF is a sufficient but not necessary condition. In this paper, a sufficient and necessary condition for G to be a differentially 4-uniform permutation is presented. We also show that G can not be an almost perfect nonlinear (APN) function. As an application, a new class of differentially 4-uniform permutations where f are not PBFs are constructed. By comparing this family with previous constructions, the number of permutations here is much bigger. The obtained functions in this paper may provide more choices for the design of Substitution boxes