Nonlinearity of vectorial functions over finite fields

The nonlinearity of a vectorial function is defined as the Hamming distance to the set of affine mappings. A connection has been established between the parameters characterizing nonlinearity and the Fourier coefficients of the characters of the vectorial function. On its basis, the possibility of finding the nonlinearity parameters of a mapping through similar parameters of its components is shown for various types of decomposition. A universal upper bound for nonlinearity is presented, expressions for the boundaries of nonlinearity are obtained in terms of the Fourier coefficients of the characters, which make it possible to clarify previously known boundaries for some classes of mappings. The dependence of the lower bound of nonlinearity on differential uniformity is found.