Constructing differential 4-uniform permutations from know ones

It is observed that exchanging two values of a function over F2n , its differential uniformity and nonlinearity change only a little. Using this idea, we find permutations of differential 4-uniform over F26 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. [1]. Moreover, for the inverse function over F2n (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 F2n .