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

2026-03-24

AI Summary

This work addresses the long-standing challenge of identifying efficiently solvable instances of the Linear Equivalence Problem (LEP), which underpins the security of several post-quantum cryptographic schemes. For the first time, the study systematically exploits algebraic structural weaknesses in LEP by introducing a novel approach that integrates power codes, Frobenius automorphisms, and Hermitian kernels. This framework generalizes Schur product–based techniques to arbitrary linear equivalence settings. By analyzing the coefficient structure within multiplicative subgroups of finite fields, the authors identify multiple new families of tractable LEP instances. Under these conditions, they achieve a significantly more efficient reduction from LEP to the Permutation Equivalence Problem (PEP), thereby substantially expanding the known solvable boundary of LEP.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Linear Equivalence Problem
• Power Codes
• Post-quantum Cryptography
• Algebraic Weaknesses
• Monomial Equivalence

Innovation

• power codes
• Linear Equivalence Problem
• Schur product
• Frobenius automorphisms
• Hermitian hulls