Characteristics of nonlinearity of vectorial functions over finite fields

Characteristics of the nonlinearity of a vectorial function defined on the vector space over a finite field are considered, namely, nonlinearity (the Hamming distance between the set of nontrivial linear combinations of its coordinate functions and the set of affine functions), differential uniformity, and another notion of nonlinearity (the Hamming distance from a vectorial function to a set of affine mappings). A method for constructing vectorial functions with high values of all these nonlinearities is demonstrated. Values of nonlinearities are found for permutations given by power functions and permutations defined by GOST R 34.11-94.