Paper 2011/047

Constructing differential 4-uniform permutations from know ones

Yuyin Yu, Mingsheng Wang, and Yongqiang Li

Abstract

It is observed that exchanging two values of a function over ${\mathbb F}_{2^n}$, its differential uniformity and nonlinearity change only a little. Using this idea, we find permutations of differential $4$-uniform over ${\mathbb F}_{2^6}$ whose number of the pairs of input and output differences with differential $4$-uniform is $54$, less than $63$, which provides a solution for an open problem proposed by Berger et al. \cite{ber}. Moreover, for the inverse function over $\mathbb{F}_{2^n}$ ($n$ even), various possible differential uniformities are completely determined after its two values are exchanged. As a consequence, we get some highly nonlinear permutations with differential uniformity $4$ which are CCZ-inequivalent to the inverse function on $\mathbb{F}_{2^n}$.