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On the Solutions of Linear Systems over Additively Idempotent Semirings

Author

Listed:
  • Álvaro Otero Sánchez

    (Department of Mathematics, University of Almería, 04120 Almería, Spain
    These authors contributed equally to this work.)

  • Daniel Camazón Portela

    (Department of Mathematics, University of Almería, 04120 Almería, Spain
    These authors contributed equally to this work.)

  • Juan Antonio López-Ramos

    (Department of Mathematics, University of Almería, 04120 Almería, Spain
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Álvaro Otero Sánchez & Daniel Camazón Portela & Juan Antonio López-Ramos, 2024. "On the Solutions of Linear Systems over Additively Idempotent Semirings," Mathematics, MDPI, vol. 12(18), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2904-:d:1480314
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