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

For a prime number p, Bergman (Israel J Math 18:257–277, 1974) established that End(\mathbbZ_p×\mathbbZ_p²){{\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 End(\mathbbZ_p ×\mathbbZ_p²){{\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 \mathbbZ_p{\mathbb{Z}_{p}} and the elements in the second row belong to \mathbbZ_p²{\mathbb{Z}_{p^{2}}}; also, using the arithmetic in \mathbbZ_p{\mathbb{Z}_{p}} and \mathbbZ_p²{\mathbb{Z}_{p^{2}}}, we introduce the arithmetic in that ring and prove that the ring End(\mathbbZ_p ×\mathbbZ_p²){{\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 \mathbbZX]{\mathbb{Z}X]}.