  In this paper we show that the $\mathbb Z/p^{m}\mathbb Z$-module structure of
the ring $E_p^{(m)}$ is isomorphic to a $\mathbb Z/p^{m}\mathbb Z$-submodule of
the matrix ring over $\mathbb Z/p^{m}\mathbb Z$. Using this intrinsic structure
of $E_p^{(m)}$, solving a linear system over $E_p^{(m)}$ becomes
computationally equivalent to solving a linear system over $\mathbb
Z/p^{m}\mathbb Z$. As an application we break the protocol based on the
Diffie-Hellman Decomposition problem and ElGamal Decomposition problem over
$E_p^{(m)}$. Our algorithm terminates in a provable running time of $O(m^{6})$
$\mathbb Z/p^{m}\mathbb Z$-operations.
