The Power of Power Codes: New Classes of Easy Instances for the Linear Equivalence Problem

TL;DR

This paper introduces new algebraic techniques using power codes, Frobenius automorphisms, and Hermitian hulls to identify easy instances of the Linear Equivalence Problem, impacting post-quantum cryptography.

Contribution

It generalizes the use of the Schur product from PEP to LEP, exploiting algebraic weaknesses to find many classes of easy-to-solve instances.

Findings

Identified new classes of easy LEP instances using algebraic methods.

Extended Schur product techniques from PEP to LEP.

Improved reduction from LEP to PEP for certain coefficient groups.

Abstract

Given two linear codes, the Linear Equivalence Problem (LEP) asks to find (if it exists) a linear isometry between them; as a special case, we have the Permutation Equivalence Problem (PEP), in which isometries must be permutations. LEP and PEP have recently gained renewed interest as the security foundations for several post-quantum schemes, including LESS. A recent paper has introduced the use of the Schur product to solve PEP, identifying many new easy-to-solve instances. In this paper, we extend this result to LEP. In particular, we generalize the approach and rely on the more general notion of power codes. Combining it with Frobenius automorphisms and Hermitian hulls, we identify many classes of easy LEP instances. To the best of our knowledge, this is the first work exploiting algebraic weaknesses for LEP. Finally we show an improved reduction to PEP whenever the coefficients of the monomial matrix are in a subgroup of the multiplicative group of the finite field.