A Note on APN Permutations and Their Derivatives

Prior to the discovery of an APN permutation in six dimension it was conjectured that such functions do not exist in even dimension, as none had been found at that time. However, finding APN permutations in even dimension ≥8 remains a significant challenge. Understanding and determining more properties of these functions is a crucial approach to discovering them. In this note, we study the properties of vectorial Boolean functions based on the weights of the first-order and second-order derivatives of their components. We show that a function is an APN permutation if and only if the sum of the squares of the weights of the first-order derivatives of its components is exactly 22n−1(2n−1+1)(2n−1). Additionally, we determined that the sum of the weights of the second-order derivatives of the components of any vectorial Boolean function is at most 22n−1(2n−1)(2n−2). This bound is achieved if and only if a function is APN.