With the advancements of the next-generation communication networking and Internet of Things (IoT) technologies, a variety of computation-intensive applications (e.g., autonomous driving and face recognition) have emerged. The execution of these IoT applications demands a lot of computing resources. Nevertheless, terminal devices (TDs) usually do not have sufficient computing resources to process these applications. Offloading IoT applications to be processed by mobile edge computing (MEC) servers with more computing resources provides a promising way to address this issue. While a significant number of works have studied task offloading, only a few of them have considered the security issue. This study investigates the problem of spectrum allocation and security-sensitive task offloading in an MEC system. Dynamic voltage scaling (DVS) technology is applied by TDs to reduce energy consumption and computing time. To guarantee data security during task offloading, we use AES cryptographic technique. The studied problem is formulated as an optimization problem and solved by our proposed efficient offloading scheme. The simulation results show that the proposed scheme can reduce system cost while guaranteeing data security.