Distance between vectorial Boolean functions and affine analogues (following the Eighth International Olympiad in Cryptography)

We study the Hamming distance from a vectorial Boolean function to a set of affine mappings (the nonlinearity of a vectorial function). New upper bound on the nonlinearity of vectorial functions and lower bound on the nonlinearity of mappings with a given differential uniformity are obtained, which refine the previously known ones. The dependence of the Hamming distance between a vectorial function and an affine mapping on the Walsh – Hadamard coefficients of nonzero linear combinations of coordinates of the vectorial function is found, which makes it possible to give estimates of nonlinearity in terms of these coefficients.