Equivalent matrices up to permutations

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.