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.

Abstract.The two important qualities of a cipher is security and speed. Frequently, to satisfy the security of a Boolean function primitive, speed may be traded-off. In this paper we present a general construction that addresses both qualities. The idea of our construction is to manipulate a cryptographically strong base function and one of its affine equivalent functions, using concatenation and negation. We achieve security from the inherent qualities of the base function, which are preserved (or increased) and obtain speed by the simple Boolean operations. We present two applications of the construction to demonstrate the flexibility and efficiency of the construction.