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

Abstract

For a prime number p , Bergman (Israel J Math 18:257–277, 1974) established that End(Zp×Zp2) is a semilocal ring with p5 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.