On the Solutions of Linear Systems over Additively Idempotent Semirings

Álvaro Otero Sánchez, Daniel Camazón Portela and Juan Antonio López-Ramos ()
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Álvaro Otero Sánchez: Department of Mathematics, University of Almería, 04120 Almería, Spain
Daniel Camazón Portela: Department of Mathematics, University of Almería, 04120 Almería, Spain
Juan Antonio López-Ramos: Department of Mathematics, University of Almería, 04120 Almería, Spain

Mathematics, 2024, vol. 12, issue 18, 1-14

Abstract: The aim of this article is to solve the system X A = Y , where A = ( a i , j ) ∈ M n × m ( S ) , Y ∈ S m and X is an unknown vector of a size n , with S being 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 with its computation. Finally, we show how to apply this method to cryptanalyze two different key exchange protocols defined for a finite case and the tropical semiring, respectively.

Keywords: linear systems over semirings; maximal solution; generalized tropical semirings; cryptography (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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