Mobile edge computing (MEC) has been envisioned as a promising scheme to address the explosive growth of computation-hungry mobile applications in future cellular systems. In this paper, we investigate the secrecy-driven energy-efficient computation offloading via MEC. Specifically, we take the secrecy-outage into account when an eavesdropper overhears the mobile terminal's (MT's) offloaded data to the edge server (ES) and formate a joint optimization of the MT's computation offloading, radio transmission, and secrecy-outage level, with the objective of minimizing the MT's energy consumption for completing its required workload. Despite the non-convexity of the joint optimization problem, we propose an efficient algorithm to find the optimal offloading solution. Based on the optimal solution for an arbitrary offloading pair of MT and ES, we further consider the scenario of multi-MT and multi-ES, and investigate the optimal pairing between the MTs and ESs for computation offloading, with the objective of minimizing all MTs' total energy consumption. Exploiting the matching structure of the pairing problem, we propose an efficient auction-based time-division scheduling algorithm to find the optimal pairing solution. Numerical results are provided to validate the effectiveness and efficiency of our proposed algorithms and the advantage of our computation offloading with secrecy-provisioning.