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

TL;DR

This paper introduces efficient characteristic set algorithms for solving polynomial equations over finite fields, with applications in cryptanalysis, including new formulas, improved algorithms, and practical implementations for Boolean polynomials.

Contribution

It presents novel algorithms for zero decomposition and characteristic sets in finite fields, including a multiplication-free method for Boolean polynomials, with proven efficiency and explicit solution formulas.

Findings

Algorithms are efficient for certain Boolean equations.

Explicit formulas for solution counts are provided.

Implementation results demonstrate practical effectiveness.

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.