Efficient Characteristic Set Algorithms for Equation Solving in Finite Fields and Applications in Cryptanalysis

Abstract

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.