Attacking the Polynomials in the Maze of Finite Fields problem

TL;DR

This paper introduces a novel, efficient method for solving a specific polynomial system over finite fields by exploiting sparsity and resultants, outperforming traditional brute-force and Gr"obner basis methods.

Contribution

The paper presents the ResultantSolver algorithm that leverages polynomial sparsity and resultants to solve finite field polynomial systems more efficiently than existing approaches.

Findings

The proposed method significantly outperforms brute-force and Gr"obner basis approaches.

Experimental results demonstrate the efficiency and effectiveness of the ResultantSolver.

Potential for parallelization suggests further performance improvements.

Abstract

In April 2025 GMV announced a competition for finding the best method to solve a particular polynomial system over a finite field. In this paper we provide a method for solving the given equation system significantly faster than what is possible by brute-force or standard Gr\"obner basis approaches. The method exploits the structured sparsity of the polynomial system to compute a univariate polynomial in the associated ideal through successive computations of resultants. A solution to the system can then be efficiently recovered from this univariate polynomial. Pseudocode is given for the proposed ResultantSolver algorithm, along with experiments and comparisons to rival methods. We also discuss further potential improvements, such as parallelizing parts of the computations.