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.

翻译：给定两个线性码，线性等价问题（LEP）要求找出（若存在）它们之间的线性等距映射；作为特例，我们考虑置换等价问题（PEP），其中等距映射必须为置换。LEP与PEP近期因作为包括LESS在内的多个后量子方案的安全基础而重新受到关注。近期一篇论文引入施尔积来求解PEP，识别出许多新型易解实例。本文将此结果推广至LEP，具体而言，我们推广了该方法，并依赖于更广义的幂码概念。结合弗罗贝尼乌斯自同构与埃尔米特对偶码，我们识别出多类易解的LEP实例。据我们所知，这是首个利用代数弱点的LEP研究工作。最后，我们证明了当单项矩阵的系数位于有限域乘法群的子群中时，可将其改进归约为PEP。