有限域方程求解的有效特征集算法及其在密码分析中的应用
Efficient Characteristic Set Algorithms for Equation Solving in Finite Fields and Applications in Cryptanalysis
摘要：
提出了一种计算有限域上多项式方程组解的有效特征集方法。引入了真三角集的概念，并给出了真一（或正则）三角集的解的个数的显式表达式。提出了一种改进的零分解算法，可将一般形式的方程组的零集约化为一元真三角集的零集的并。因此，我们可以给出一个方程组的解的数目的显式公式。在布尔多项式的情况下，给出了算法的比特复杂度。我们还给出了布尔多项式的一个无乘法特征集方法，其中多项式的大小得到了有效的控制。算法的情况下，布尔多项式和广泛的实验表明，他们是相当有效的解决某些类别的布尔方程。
英文摘要：
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper and monic (or regular) triangular set is given. An improved zero decomposition algorithm which can be used to reduce the zero set of an equation system in general form to the union of zero sets of monic proper triangular sets is proposed. As a consequence, we can give an explicit formula for the number of solutions of an equation system. Bitsize complexity for the algorithm is given in the case of Boolean polynomials. We also give a multiplication free characteristic set method for Boolean polynomials, where the sizes of the polynomials are effectively controlled. The algorithms are implemented in the case of Boolean polynomials and extensive experiments show that they are quite efficient for solving certain classes of Boolean equations.
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