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

Abstract

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 ${\mathbb{Z}[X]}$.