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Mobile edge computing can provide powerful computation services around the end-users. However, given the broadcast nature of wireless transmissions, offloading the computation tasks via the uplink channels would raise serious security concerns. This paper proposes an online approach to jointly optimize local processing, transmit power, and task offloading decisions without the a-priori knowledge of the dynamic environments. The proposed approach can guarantee the secure offloading and asymptotically minimize the time-average energy consumption of devices while maintaining the stability of the ergodic secrecy queues and task queues. By exploiting the Lyapunov optimization, the local processing, transmit power, and task offloading variables can be decoupled between time slots. The subproblems on local processing and computation offloading can be solved separately. Convex optimization and graph matching can be used to solve the computation offloading subproblem. Simulations show that the performances of the proposed approach are superior to other popular approaches.