The aim of this article is to solve the system $XA=Y$ where $A=(a_{ij})\in M_{m\times n}(S)$, $Y\in S^{m}$ and $X$ is an unknown vector of size $n$, being $S$ an additively idempotent semiring. If the system has solutions then we completely characterize its maximal one, and in the particular case where $S$ is a generalized tropical semiring a complete characterization of its solutions is provided as well as an explicit bound of the computational cost associated to its computation. Finally, when $S$ is finite, we give a cryptographic application by presenting an attack to the key exchange protocol proposed by Maze, Monico and Rosenthal.

翻译：本文旨在求解系统 $XA=Y$，其中 $A=(a_{ij})\in M_{m\times n}(S)$，$Y\in S^{m}$，$X$ 为大小为 $n$ 的未知向量，$S$ 是加法幂等半环。若该系统存在解，则我们完全刻画其最大解；在 $S$ 为广义热带半环的特殊情形下，我们进一步给出所有解的完整刻画，并明确其计算过程的计算代价上界。最后，当 $S$ 有限时，我们提出一种密码学应用，即对 Maze、Monico 和 Rosenthal 提出的密钥交换协议实施攻击。