Analyzing Boolean Functions via Solving Parametric Polynomial Systems

Abstract

In this paper, a new method to analyze Boolean functions is proposed. By this method, one can analyze the balancedness, the nonlinearity, and the input-output correlation of vectorial Boolean functions. The basic idea of this method is to compute the refined covers of some parametric Boolean polynomial systems which are equivalent to these problems. By a refined cover, the parameter space is divided into several disjoint components, and on each component, the parametric Boolean polynomial system has a fixed number of solutions. An efficient algorithm based on the characteristic set method to compute refined covers of parametric Boolean polynomial systems is presented. The experimental results about some instances generated from cryptanalysis show that this new method is efficient and can solve some instances which can not be solved in reasonable time by other methods.