Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Michele Battagliola, Anna-Lena Horlemann, Abhinaba Mazumder, Rocco Mora, Paolo Santini, Michael Schaller, Violetta Weger

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.

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

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.
Paper Structure (11 sections, 5 theorems, 32 equations, 2 tables, 1 algorithm)

This paper contains 11 sections, 5 theorems, 32 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Coding Theory Basics
  5. The Code Equivalence Problem
  6. Power Codes
  7. Odd $q$
  8. Frobenius automorphisms for non-prime fields
  9. Hermitian hulls and further
  10. Odd degree extensions
  11. Partial Closure

Key Result

Theorem 6

hull Let $\mathcal{A}, \mathcal{B}$ be two $[n,k]_q$ linear codes. Then $\mathcal{A}$ is linearly equivalent to $\mathcal{B}$ if and only if $\boldsymbol{\alpha} \otimes \mathcal{A}$ is permutation equivalent to $\boldsymbol{\alpha} \otimes \mathcal{B}$.

Theorems & Definitions (14)

  • Definition 1: Hull
  • Definition 2: Power Code
  • Definition 3: Hermitian Inner Product
  • Definition 4: Closure
  • Definition 5: Linear Equivalence Problem (LEP)
  • Theorem 6: Reducing LEP to PEP
  • Proposition 7
  • Proposition 8
  • proof
  • Theorem 9
  • ...and 4 more