Attacking the Polynomials in the Maze of Finite Fields problem

TL;DR

This paper provides a method for solving the given equation system significantly faster than what is possible by brute-force or standard Grobner basis approaches.

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 Gröbner 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.

Figures (3)

• Figure 1: Binary tree describing the algorithm from Section 3, with $n=7$.
• Figure 2: Binary tree describing the iterative pairwise computing of the polynomials $f_{[i+1, j]}\in\mathbb{F}_p[x_1, x_{i}, x_j,x_n]$. See Example \ref{['ex:n7']} for explicit description for the first resultants. The notation $f(x_{a\sout{b}c})$ means that the polynomial $f$ contains variables $x_a,x_b,x_c$ and the variable $x_b$ is eliminated in the next step.
• Figure 3: Final steps of the algorithm, when $n=7$. Notice that only this part of the algorithm depends on $f_7$, and thereby on the parameter $t$. The value $x_n$ for a solution to the polynomial $u$ is the desired output.

Theorems & Definitions (8)

• Proposition 1
• Corollary 1
• Proposition 2
• Remark 1
• Proposition 3
• Proposition 4
• Conjecture 1
• Example 1