Nonlinearity of APN functions: comparative analysis and estimates

The main results of the paper relate to the nonlinearity of APN functions defined for a vectorial Boolean function as the Hamming distance from it to the set of affine mappings in the space of images of all vectorial Boolean functions in fixed dimension. For APN functions in dimension $n$, the lower nonlinearity bound of the form $2^n - \sqrt {2^{n+1} - 7\cdot2^{-2}} - 2^{-1}$ and the corresponding lower bound on the affinity order are obtained. The exact values of the nonlinearity of all APN functions up to dimension $5$ are found, and also for one known APN $6$-dimensional permutation and for all differentially $4$-uniform permutations in dimension $4$.