In recent years, edge computing has attracted significant attention because it can effectively support many delay-sensitive applications. Despite such a salient feature, edge computing also faces many challenges, especially for efficiency and security, because edge devices are usually heterogeneous and may be untrustworthy. To address these challenges, we propose a unified framework to provide efficiency and confidentiality by coded distributed computing. Within the proposed framework, we use matrix multiplication, a fundamental building block of many distributed machine learning algorithms, as the representative computation task. To minimize resource consumption while achieving information-theoretic security, we investigate two highly-coupled problems, (1) task allocation that assigns data blocks in a computing task to edge devices and (2) linear code design that generates data blocks by encoding the original data with random information. Specifically, we first theoretically analyze the necessary conditions for the optimal solution. Based on the theoretical analysis, we develop an efficient task allocation algorithm to obtain a set of selected edge devices and the number of coded vectors allocated to them. Using the task allocation results, we then design secure coded computing schemes, for two cases, (1) with redundant computation and (2) without redundant computation, all of which satisfy the availability and security conditions. Moreover, we also theoretically analyze the optimization of the proposed scheme. Finally, we conduct extensive simulation experiments to demonstrate the effectiveness of the proposed schemes.