This paper investigates a multi-user uplink mobile edge computing (MEC) network, where the users offload partial tasks securely to an access point under the non-orthogonal multiple access policy with the aid of a reconfigurable intelligent surface (RIS) against a multi-antenna eavesdropper. We formulate a non-convex optimization problem of minimizing the total energy consumption subject to secure offloading requirement, and we build an efficient block coordinate descent framework to iteratively optimize the number of local computation bits and transmit power at the users, the RIS phase shifts, and the multi-user detection matrix at the access point. Specifically, we successively adopt successive convex approximation, semi-definite programming, and semidefinite relaxation to solve the problem with perfect eavesdropper's channel state information (CSI), and we then employ S-procedure and penalty convex-concave to achieve robust design for the imperfect CSI case. We provide extensive numerical results to validate the convergence and effectiveness of the proposed algorithms. We demonstrate that RIS plays a significant role in realizing a secure and energy-efficient MEC network, and deploying a well-designed RIS can save energy consumption by up to 60\% compared to that without RIS. We further reveal impacts of various key factors on the secrecy energy efficiency, including RIS element number and deployment position, user number, task scale and duration, and CSI imperfection.

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