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)要求找到（若存在）它们之间的线性等距映射；作为特例，我们有置换等价问题(PEP)，其中等距映射必须是置换。LEP和PEP作为多个后量子方案（包括LESS）的安全基础，近来重新引起关注。最近一篇论文引入了利用Schur积求解PEP的方法，识别出许多新的易解实例。在本文中，我们将此结果推广到LEP。具体而言，我们推广了该方法，并依赖于更一般的幂码概念。结合Frobenius自同构和Hermitian hull，我们识别出多类易解的LEP实例。据我们所知，这是首篇利用代数弱点处理LEP的工作。最后，我们展示了当单项矩阵的系数位于有限域乘法群的子群中时，向PEP的改进归约。