Solving Linear Systems over Additively Idempotent Semirings: Characterization of Maximal Solutions and Computational Complexity

Core Concepts

The core message of this article is to provide a complete characterization of the maximal solution of linear systems over additively idempotent semirings, and to analyze the computational complexity of solving such systems, particularly in the case of generalized tropical semirings.

Abstract

The article starts by introducing the concept of additively idempotent semirings and their properties. It then focuses on solving linear systems of the form XA = Y, where A is a matrix with entries in an additively idempotent semiring S, Y is a vector in S, and X is the unknown vector.
The key results are:

For general additively idempotent semirings, the authors show that if the system has solutions, then the maximal solution can be completely characterized using the sets Wi = {x ∈ S : xAi + Y = Y}, where Ai is the i-th row of A.

For the case where S is a generalized tropical semiring (where a + b = a or a + b = b for all a, b ∈ S), the authors provide a complete characterization of the solutions and an explicit bound on the computational cost of finding them.

For the case where S is a finite additively idempotent semiring, the authors show that the system is always compatible and the maximal solution can be computed efficiently.

The authors also provide a cryptographic application of their results, showing how the solution of linear systems over finite semirings can be used to attack a key exchange protocol.

The article presents a comprehensive analysis of linear systems over additively idempotent semirings, with both theoretical and practical implications.

Stats

The computational cost of solving linear systems over generalized tropical semirings is O(nm), where n is the number of rows in the coefficient matrix A and m is the number of columns.

Quotes

"Let (R, +, ·) be a generalized tropical semiring such that (R, ·) is a group and let XA = Y be a system of equations with Y ∈ Rm and A = (ai,j) ∈ Matn×m(R). Determining all the solutions of the system has a computational cost of O(nm)."

Key Insights Distilled From

On the solutions of linear systems over additively idempotent semirings

by Álva... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03294.pdf

Deeper Inquiries

How can the results in this article be extended to other classes of semirings beyond the generalized tropical semirings

The results in the article can be extended to other classes of semirings beyond generalized tropical semirings by considering different algebraic structures and properties. For example, one could explore the solutions of linear systems over semirings with specific properties such as idempotent elements, cancellative elements, or distributive laws. By adapting the algorithms and characterizations developed for generalized tropical semirings to these different classes of semirings, one can potentially solve linear systems and characterize their solutions in various algebraic settings.

What are the potential applications of solving linear systems over additively idempotent semirings in areas such as discrete optimization, fuzzy logic, or automata theory

Solving linear systems over additively idempotent semirings has numerous potential applications in various fields. In discrete optimization, these solutions can be used to model and solve optimization problems efficiently, especially when the problem can be linearized over an additively idempotent semiring. In fuzzy logic, the solutions of linear systems can help in reasoning and decision-making processes where uncertainty and imprecision are present. Additionally, in automata theory, these solutions can be applied to model and analyze complex systems with discrete states and transitions, providing insights into system behavior and properties.

Can the cryptographic attack presented in the article be generalized to other key exchange protocols based on linear systems over finite semirings

The cryptographic attack presented in the article, which involves solving a linear system over a finite semiring to break a key exchange protocol, can potentially be generalized to other key exchange protocols based on linear systems over finite semirings. By identifying vulnerabilities in the key exchange process that can be exploited through solving linear systems, similar attacks could be devised for other protocols. However, the applicability of the attack would depend on the specific properties and structures of the semiring used in the key exchange protocol, as well as the encryption and decryption mechanisms employed. Further research and analysis would be needed to assess the feasibility and effectiveness of such attacks in different cryptographic scenarios.