A new family of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2k} }$$ for odd k

Abstract

We study the differential uniformity of a class of permutations over $\mathbb{F}_{2^{n}}$ with $n$ even. These permutations are different from the inverse function as the values $x^{-1}$ are modified to be $(\gamma x)^{-1}$ on some cosets of a fixed subgroup $\langle\gamma\rangle$ of $\mathbb{F}_{2^{n}}^*$. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) classes as checked by Magma for small numbers $n$. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.