A formula for systems of Boolean polynomial equations and applications to computational complexity

Abstract

It is known a method for transforming a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we improve the method, and show a formula in the Boolean polynomial ring for systems of Boolean polynomial equations. The formula has both operations of conjunction and disjunction recursively, and has a structure of binary tree.As corollaries, we prove computational complexity results for solving a given system. The complexity results include parameters similar to the bandwidth and rank of the matrix, and they are influenced by the order of variables in Boolean polynomials forming the system. Comparisons with existing complexity results of NP-complete problems related to systems of Boolean polynomial equations are also mentioned.