On the behavior of some APN permutations under swapping points

Abstract

We define the pAPN-spectrum (which is a measure of how close a function is to being APN) of an (n, n)-function F and investigate how its size changes when two of the outputs of a given function F are swapped. We completely characterize the behavior of the pAPN-spectrum under swapping outputs when F is the inverse function over F2n. We further theoretically investigate this behavior for functions from the Gold and Welch monomial APN families, and experimentally determine the size of the pAPN-spectrum after swapping outputs for representatives from all infinite monomial APN families up to dimension n = 10; based on our computation results, we conjecture that the inverse function is the only monomial APN function for which swapping two of its outputs can leave an empty pAPN-spectrum.