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

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 (γx)^? on some cosets of a fixed subgroup 〈γ〉 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.