The number of affine equivalent classes and extended affine equivalent classes of vectorial Boolean functions

Abstract

Affine equivalent classes and extended affine equivalent (EA-equivalent for short) classes of vectorial Boolean functions have important applications in cryptography, logic circuit, sequences for communications, etc. Recently, Y. Zhang et al., computed the number of affine equivalent classes of n-variable Boolean functions when 1≤n≤10 by group isomorphism (Zhang et al., 2016). However, the case for affine equivalent vectorial Boolean function remains a challenging open problem. Furthermore, little result for EA-equivalent vectorial Boolean function is known except a trivial lower bound mentioned in Mullen and Panario (2013, P246). In this paper, we focus on the challenging problem of calculating the number of affine equivalent classes and EA-equivalent classes of vectorial Boolean functions. First, for EA-equivalence, we prove that the trivial lower bound proposed in Mullen and Panario (2013) has at least 3 effective figures if n is not too small. We then show that the lower bound also holds for affine equivalent classes. Furthermore, we give an explicit formula and calculate the exact number of affine equivalent classes of (n,m)-functions with 1≤m,n≤11 by GAP. The results in this paper are helpful for the theory and applications of the classifications of vectorial Boolean functions.