In this paper, we consider a mobile edge computing(MEC)-based secure computation offloading system, and design a practical multi-eavesdropper model including two specific scenarios of non-colluding and colluding eavesdropping. Furthermore, we design a requirement satisfaction model by exploring practical variations in user request patterns for security provisioning, delay reduction and energy saving. Based on these, we propose a satisfaction-maximized secure computation offloading (SMaxSCO) scheme, and then formulate an optimization problem aiming at maximizing users’ requirement satisfactions subject to secrecy offloading rate, tolerable delay, task workload and maximum power constraints. Since the optimization problem is nonconvex, we present an efficient successive convex approximation (SCA)-based algorithm to obtain suboptimal solutions. We demonstrate that the proposed SMax-SCO scheme achieves a significant improvement in security performance and requirement satisfaction compared with existing schemes. Moreover, we conclude that SMax-SCO can resist eavesdropping attacks of multiple eavesdroppers and even colluding eavesdroppers.