On the solutions of linear systems over additively idempotent semirings

TL;DR

This work analyzes solving linear systems $X A = Y$ over additively idempotent semirings, providing a complete characterization of the maximal solution and, in the generalized tropical setting, a full solution with explicit computational cost bounds. It develops a robust framework based on the sets $W_i$ and their maxima $C_i$, and extends results to tropical and generalized tropical semirings, including embedding techniques when inverses are not available. In the finite case, the maximal solution is obtained by summing the finite sets $W_i$, and the authors apply these results to cryptography by showing how solving associated linear systems can break a finite semiring-based key-exchange protocol. The findings advance linear algebra over idempotent semirings, with implications for tropical algebra, discrete optimization, and cryptographic vulnerabilities in semiring-based schemes.

Abstract

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.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Definition 3.1
• Lemma 3.2
• Example 3.3