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

Abstract

For a prime number $$p$$ p , Bergman (Israel J Math 18:257---277, 1974) established that $$\mathrm {End}(\mathbb {Z}_{p} \times \mathbb {Z}_{p^{2}})$$ End ( Z p × Z p 2 ) is a semilocal ring with $$p^{5}$$ 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.