Mobile edge computing (MEC) has been envisaged as a promising technique in the next-generation wireless networks. In order to improve the security of computation tasks offloading and enhance user connectivity, physical layer security and non-orthogonal multiple access (NOMA) are studied in MEC-aware networks. The secrecy outage probability is adopted to measure the secrecy performance of computation offloading by considering a practically passive eavesdropping scenario. The weighted sum-energy consumption minimization problem is firstly investigated subject to the secrecy offloading rate constraints, the computation latency constraints and the secrecy outage probability constraints. The semi-closed form expression for the optimal solution is derived. We then investigate the secrecy outage probability minimization problem by taking the priority of two users into account, and characterize the optimal secrecy offloading rates and power allocations with closed-form expressions. Numerical results demonstrate that the performance of our proposed design are better than those of the alternative benchmark schemes.Index Terms-Mobile edge computing, non-orthogonal multiple access, physical layer security, secrecy outage probability, partial offloading.