Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

Abstract

a b s t r a c tThe 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 < p 2 . Also, we find a connection between the generalized inverse of a circulant matrix and the invertibility of its generating polynomial over F 2 , modulo a product of cyclotomic polynomials, thus generalizing a known result on nonsingular circulant matrices.Published by Elsevier B.V.

Counting equivalence classes for monomial rotation symmetric Boolean functions with prime dimension

“…There are 5 other standard forms, in which the variables in the first monomial [3, 2, 1] above are permuted; the most natural of the 6 standard forms would begin with the monomial [1,2,3]. Note the 1-terms of this function are [1,2,3], [1,2,4] and [1,3,4].…”

Counting permutation equivalent degree six binary polynomials invariant under the cyclic group

“…The main result of this paper is to find the exact number of equivalence classes (and representatives of these classes) for sextic (degree 6) MRS (whose SANF is f = x 1 x i x j x k x s x t with ( f ) = {1, i, j, k, s, t}) in prime dimensions; the cubic, quartic and quintic cases were done previously in [2,[4][5][6][7]9]; in [11], one of us completely solved the case of quartics in prime power dimension.…”

“…We recall now (see [2]) another type of equivalence between circulant matrices and their equivalence classes. Two circulant matrices A, B are called P-Q equivalent, if P B = AQ, where P, Q are permutation matrices.…”

“…are P-equivalent [2] and written f P ∼ g, if f, g are affine equivalent under a permutation of variables (we often write f ∼ g, or,…”

“…, c n ), if all its rows are successive (right) circular rotations of the first row. On the set C n of circulant matrices an equivalence relation was introduced in [2]: for…”