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 $${{\rm End}(\mathbb{Z}_{p}\times \mathbb{Z}_{p^{2}})}$$ is a semilocal ring with p5 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 Ep, 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 Ep. Finally, we present a Diffie-Hellman key interchange protocol using some polynomial functions over Ep defined by polynomials in $${\mathbb{Z}[X]}$$.