An extension of the noncommutative Bergman’s ring with a large number of noninvertible elements

  • Joan-Josep ClimentEmail author
  • Pedro R. Navarro
  • Leandro Tortosa
Original Paper


For a prime number \(p\), Bergman (Israel J Math 18:257–277, 1974) established that \(\mathrm {End}(\mathbb {Z}_{p} \times \mathbb {Z}_{p^{2}})\) is a semilocal ring with \(p^{5}\) elements that cannot be embedded in matrices over any commutative ring. In an earlier paper Climent et al. (Appl Algebra Eng Commun Comput 22(2):91–108, 2011), the authors presented an efficient implementation of this ring, and introduced a key exchange protocol based on it. This protocol was cryptanalyzed by Kamal and Youssef (Appl Algebra Eng Commun Comput 23(3–4):143–149, 2012) using the invertibility of most elements in this ring. In this paper we introduce an extension of Bergman’s ring, in which only a negligible fraction of elements are invertible, and propose to consider a key exchange protocol over this ring.


Noncommutative ring Noninvertible element Key exchange protocol Endomorphism Bergman ring 



The authors are very grateful to the anonymous reviewers for their comments and suggestions which led to significant improvements.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Joan-Josep Climent
    • 1
    Email author
  • Pedro R. Navarro
    • 2
  • Leandro Tortosa
    • 2
  1. 1.Departament d’Estadística i Investigació OperativaUniversitat d’AlacantAlacantSpain
  2. 2.Departament de Ciència de la Computació i Intel∙ligència ArtificialUniversitat d’AlacantAlacantSpain

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