ON THE ARITHMETIC OF ENDOMORPHISM RING End$(\mathbb{Z}_{p^2} \times \mathbb{Z}_{p})$ AND ITS RSA VARIANTS

Bergman (1974) found that for any prime number $p$, the endomorphism ring End$(\mathbb{Z}_{p} \times \mathbb{Z}_{p^2})$ is a semilocal ring which has $p^5$ elements and can not be embedded in matrices over any commutative ring. Later on, Climent et al. (2011) found that each element of endomorphism ring End$(\mathbb{Z}_{p} \times \mathbb{Z}_{p^2})$ can be identified as a two by two matrix of $E_p$ where the first and the second row entries belong to $\mathbb{Z}_{p}$ and $\mathbb{Z}_{p^2}$ respectively. By this characterization, Long D.T., Thu D. T., and Thuc D. N. constructed a new RSA variant based on End$(\mathbb{Z}_{p} \times \mathbb{Z}_{p^2})$ (2013). In this paper, we state the characteristic of the endomorphism ring End $(\mathbb{Z}_{p^2} \times \mathbb{Z}_{p})$ and the RSA analogue cryptosystem based on it.