Abstract
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.
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The authors are very grateful to the anonymous reviewers for their comments and suggestions which led to significant improvements.
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The work of this author was partially supported by Spanish Grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España.
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Climent, JJ., Navarro, P.R. & Tortosa, L. An extension of the noncommutative Bergman’s ring with a large number of noninvertible elements. AAECC 25, 347–361 (2014). https://doi.org/10.1007/s00200-014-0231-6
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DOI: https://doi.org/10.1007/s00200-014-0231-6