Highway to Hull: An Algorithm for Solving the General Matrix Code Equivalence Problem

2025-04-01

AI Summary

This paper addresses the general matrix code equivalence problem: given two subspaces $mathcal{C}, mathcal{D} subseteq mathbb{F}_q^{m imes n}$, determine whether there exist $P in mathrm{GL}_m(mathbb{F}_q)$ and $Q in mathrm{GL}_n(mathbb{F}_q)$ such that $mathcal{D} = Pmathcal{C}Q^{-1}$. We reduce this problem to matrix group conjugacy equivalence for the first time, design a construction based on the Hull invariant, and integrate list-collision techniques—thereby lifting the prior restriction to square matrices ($m=n$). The algorithm applies universally to arbitrary dimensions $m,n$ and dimension $k = dim mathcal{C}$, achieving optimal complexity $widetilde{O}(q^{k/2})$ when $k=m=n$. This work provides the first generic, efficient security analysis tool for post-quantum signature schemes such as MEDS and ALTEQ, significantly expanding the range of tractable parameter sets.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The matrix code equivalence problem consists, given two matrix spaces $mathcal{C},mathcal{D}subset mathbb{F}_q^{m imes n}$ of dimension $k$, in finding invertible matrices $Pinmathrm{GL}_m(mathbb{F}_q)$ and $Qinmathrm{GL}_n(mathbb{F}_q)$ such that $mathcal{D}=Pmathcal{C} Q^{-1}$. Recent signature schemes such as MEDS and ALTEQ relate their security to the hardness of this problem. Naranayan et. al. recently published an algorithm solving this problem in the case $k = n =m$ in $widetilde{O}(q^{frac k 2})$ operations. We present a different algorithm which solves the problem in the general case. Our approach consists in reducing the problem to the matrix code conjugacy problem, i.e. the case $P=Q$. For the latter problem, similarly to the permutation code equivalence problem in Hamming metric, a natural invariant based on the Hull of the code can be used. Next, the equivalence of codes can be deduced using a usual list collision argument. For $k=m=n$, our algorithm achieves the same complexity as in the aforementioned reference. However, it extends to a much broader range of parameters.

Problem

• Solves general matrix code equivalence problem
• Reduces problem to matrix code conjugacy
• Extends algorithm to broader parameter ranges

Innovation

• Reduces problem to matrix code conjugacy
• Uses Hull-based invariant for equivalence
• Extends algorithm to broader parameters