Constructing new differentially 4-uniform permutations from the inverse function

Two new families of differentially 4-uniform permutations over F22mF22m are constructed by modifying the values of the inverse function on some subfield of F22mF22m and by applying affine transformations on the function. The resulted 4-uniform permutations have high nonlinearity and algebraic degree. A family of differentially 6-uniform permutations with high nonlinearity and algebraic degree is also constructed by making the modification on an affine subspace of F22mF22m.