Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

Abstract

The goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám’s conjecture from graph theory. As applications, we reprove easily several main results of Cusick et al. on the number of equivalence classes under permutations for MRS in prime power dimensions, as well as give a count for the number of classes in pq number of variables, where p,q are prime numbers with p<q<p2. Also, we find a connection between the generalized inverse of a circulant matrix and the invertibility of its generating polynomial over F2, modulo a product of cyclotomic polynomials, thus generalizing a known result on nonsingular circulant matrices.