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In recent years, edge computing (EC) has attracted great attention for its high-speed computing and low-latency characteristics. However, there are many challenges in the implementation of EC. Firstly, user's privacy has been raised as a major concern because the edge devices may be untrustworthy. In the case of Private Edge Computing (PEC), a user wants to compute a matrix multiplication between its local matrix and one of the matrices in a library, which has been redundantly stored in edge devices. When utilizing resources of edge devices, the privacy requires that each edge device cannot know which matrix stored on it is desired by the user for the multiplication. Secondly, edge devices usually have limited communication and storage resources, which makes it impossible for them to store all matrices in the library. In this paper, we consider the limited resources of edge devices and propose an unified framework for PEC. Within the framework, we study two highly-coupled problems, (1) storage allocation, that determines which matrices are stored on each edge device, and (2) computation design, that determines which matrices (or linear combinations of them) in each edge device are selected to participate in the computing process with the privacy consideration. Specifically, we give a general storage allocation scheme and then design two feasible private computation schemes, i.e., General Private Computation (GPC) scheme and Private Coded Computation (PCC) scheme. In particular, GPC can be applied in general case and PCC can only be applied in special cases, while PCC achieves less communication load. We theoretically analyze the proposed computing schemes and compare them with other schemes. Finally, we conduct extensive simulations to show the effectiveness of the proposed schemes.