On the arithmetic of the endomorphisms ring \({{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}\)

  • 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 \({{\rm End}(\mathbb{Z}_{p}\times \mathbb{Z}_{p^{2}})}\) is a semilocal ring with p 5 elements that cannot be embedded in matrices over any commutative ring. We identify the elements of \({{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}\) with elements in a new set, denoted by E p , of matrices of size 2 × 2, whose elements in the first row belong to \({\mathbb{Z}_{p}}\) and the elements in the second row belong to \({\mathbb{Z}_{p^{2}}}\); also, using the arithmetic in \({\mathbb{Z}_{p}}\) and \({\mathbb{Z}_{p^{2}}}\), we introduce the arithmetic in that ring and prove that the ring \({{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}\) is isomorphic to the ring E p . Finally, we present a Diffie-Hellman key interchange protocol using some polynomial functions over E p defined by polynomials in \({\mathbb{Z}[X]}\).


Endomorphism Isomorphism Noncommutative ring Additive order Invertible element Key exchange protocol 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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

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