On the Annihilators of Boolean Polynomials

Abstract

Boolean functions in general and Boolean polynomials (Zhegalkin polynomials or algebraic normal forms (ANF)) in particular are the subject of theoretical and applied studies in various fields of computer science. This article addresses the linear operators of the space of Boolean polynomials in n variables, which leads to the results on the problem of finding the minimum annihilator degree for a given Boolean polynomial. This problem is topical in various analytical and algorithmic aspects of cryptography. Boolean polynomials and their combinatorial properties are under study in discrete analysis. The theoretical foundations of information security include the study of the properties of Boolean polynomials in connection with cryptography. In this article, we prove a theorem on the minimum annihilator degree. The class of Boolean polynomials is described for which the degree of an annihilator is at most 1. We give a few combinatorial characteristics related to the properties of the space of Boolean polynomials. Some estimates of the minimum degree of an annihilator are given. We also consider the case of symmetric polynomials.