In this paper, we focus on the efficient offloading mechanism in multi-user mobile edge computing (MEC) system, where an MEC server is supposed to serve K(> 1) mobile devices with different computation load and priority in the presence of one eavesdropper. The factors that affect the communication and computation resource allocation of the MEC server to different users will be addressed to highlight the tradeoff between execution latency and energy consumption in multi-user MEC system. In order to reveal the influence of relative channel conditions, different computation loads, and different priorities, a weighted summation optimization problem of energy consumption and execution latency was formulated. Because the weighted summation optimization problem is non-convex and difficult to solve, Lagrange dual decomposition was utilized to decompose the original optimization problem into multiple subproblems, which can be processed separately. Because the subproblems are still nonconvex, we further decompose the subproblems into two-step optimization problem, which can be solved by using Karush-Kuhn-Tucker conditions and precise line search algorithm. Our analysis results show that, the proposed secure offloaing design can ensure those mobile devices with more computing tasks and higher priority level have precedence in offloading computation tasks to MEC servers in the presence of eavesdropper.