New explicit constructions of differentially 4-uniform permutations via special partitions of F22kF22k

In this paper, we further study the switching constructions of differentially 4-uniform permutations over F22kF22k from the inverse function and propose several new explicit constructions. In our constructions, we first partition the finite field F22kF22k into some minimal subsets that are closed under both mappings x↦1x−1+1 and x↦ωxx↦ωx, where ω∈F22k⁎ is of order 3. Then, by utilizing some properties of such subsets to extend differentially 4-uniform permutations over the subfield F4F4 or F24F24 to that over F22kF22k, we give new constructions of differentially 4-uniform permutations over F22kF22k for the cases k   odd, k/2k/2 odd and k/2k/2 even respectively. As compared to previous constructions, our new constructions explicitly give large numbers (at least 222k−2−1222k−2−1) of functions.