Equivalent matrices up to permutations

TL;DR

This paper explores algebraic geometric methods to solve matrix equations involving permutation matrices, with applications to cryptanalysis of McEliece cryptosystems based on Reed-Solomon codes.

Contribution

It introduces algebraic geometric techniques for solving matrix equations with permutation matrices and applies them to efficiently break McEliece cryptosystems.

Findings

Methods successfully solve matrix equations involving permutations.

Application to cryptanalysis yields efficient key recovery.

Potential impact on cryptographic security assessments.

Abstract

Given two $k\times n$ matrices $A$ and $B$, we describe a couple of methods to solve the matrix equation $XA=BY$, where $X$ is an invertible $k\times k$ matrix, and $Y$ is an $n\times n$ permutation matrix, both of which we want to determine. We are interested in pursuing those techniques that have algebraic geometric flavor. An application to solving such a matrix equation comes from the cryptanalysis of McEliece cryptosystem. By using codewords of minimum weight of a linear code, in concordance with these methods of solving $XA=BY$, we present an efficient way to determine the entire encryption keys for the McEliece cryptosystems built on Reed-Solomon codes.