The assumed hardness of the Linear Code Equivalence problem (LCE) lies at the core of the security of the LESS signature scheme and other signature schemes with advanced functionalities. The LCE problem asks to determine whether two linear codes are equivalent. This equivalence is represented by a monomial matrix $ Q$, i.e. the product of a diagonal matrix $D$ and a permutation matrix $P$. The recovery of $Q=DP$ is known to be reduced to the recovery of the permutation matrix $ P$ alone. Exploiting this fact, we construct an algebraic model for LCE involving only the matrix $P$. To this end, we study the action of monomial matrices on linear codes using tools from algebraic geometry, including Plücker coordinates and fields of invariant rational functions. In particular, we analyse the action of diagonal matrices on linear codes, which can be interpreted as diagonal scaling of the coordinates of elements of the Grassmannian. We propose a method to determine algebraically independent generators of the field of rational functions invariant under this action, without relying on Reynolds operators or Gröbner basis computations. Furthermore, given two equivalent codes, we apply our results to explicitly construct, for each invariant function, a polynomial having $P$ as a root. However, the resulting polynomials are not of practical use: their degrees are high for cryptographically relevant parameters, and the number of monomials grows exponentially, making them infeasible to manipulate. Despite this limitation, our results are of theoretical interest, as they constitute the first application of these tools to the cryptanalysis of LCE and provide insight into how algebraic geometry and invariant theory can be employed in Cryptography.

翻译：线性码等价问题（LCE）的假设困难性是LESS签名方案及其他具有高级功能的签名方案安全性的核心基础。LCE问题要求判定两个线性码是否等价，这种等价性由单项矩阵$Q$表示，即对角矩阵$D$与置换矩阵$P$的乘积。已知恢复$Q=DP$可简化为仅恢复置换矩阵$P$。利用这一事实，我们构建了一个仅涉及矩阵$P$的LCE代数模型。为此，我们运用代数几何工具（包括普吕克坐标与不变有理函数域）研究单项矩阵在线性码上的作用。特别地，我们分析了对角矩阵在线性码上的作用，该作用可理解为格拉斯曼流形元素坐标的对角缩放。我们提出了一种方法，可在不依赖Reynolds算子或Gröbner基计算的情况下，确定在该作用下不变的有理函数域的代数独立生成元。此外，给定两个等价码，我们应用研究结果显式构造了每个不变函数对应的、以$P$为根的多项式。然而，所得多项式缺乏实用价值：对于密码学相关参数，其次数过高，且单项式数量呈指数增长，导致无法实际运算。尽管存在这一局限，我们的结果具有理论意义——这是首次将这些工具应用于LCE的密码分析，并揭示了代数几何与不变量理论在密码学中的运用途径。