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.

本文提出了一种分析布尔函数的新方法。通过这种方法，可以分析矢量布尔函数的平衡性，非线性和输入输出相关性。此方法的基本思想是计算一些等效于这些问题的参数布尔多项式系统的精确覆盖。通过精巧的覆盖，参数空间被分成几个不相交的分量，并且在每个分量上，参数布尔多项式系统都有固定数量的解。提出了一种基于特征集方法计算参数布尔多项式系统精巧覆盖的有效算法。