Abstract
In this paper we show that the Z/p^mZ-module structure of the ring E_p^(m) is isomorphic to a Z/p^mZ-submodule of the matrix ring over Z/p^mZ. 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 Z/p^mZ. 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(m⁶) Z/p^mZ-operations.