Solving one-sided linear systems over symmetrized and supertropical semiring

TL;DR

This paper extends methods for solving one-sided linear systems from tropical semirings to symmetrized and supertropical semirings, with implications for tropical cryptography.

Contribution

It introduces an approach to solve such systems over symmetrized and supertropical semirings, expanding the applicability of tropical linear algebra.

Findings

Efficiently finds the greatest solution over extended semirings.

Addresses the complexity of finding all minimal solutions.

Discusses implications for tropical cryptography.

Abstract

One-sided linear systems of the form ``$Ax=b$'' are well-known and extensively studied over the tropical (max-plus) semiring and wide classes of related idempotent semirings. The usual approach is to first find the greatest solution to such system in polynomial time and then to solve a much harder problem of finding all minimal solutions. We develop an extension of this approach to the same systems over two well-known extensions of the tropical semiring: symmetrized and supertropical, and discuss the implications of our findings for the tropical cryptography.