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Mobile devices (MDs) and applications are receiving extensive popularity and attracting significant attention. Mobile applications, especially for artificial intelligence (AI) applications, require powerful computation-intensive resources. Hence, running all the AI applications on a single MD introduces high energy consumption and application delay, as it has limited battery capacity and computation resources. Fortunately, the emerging edge-cloud computing (ECC) architecture pushes the computation resource to both the network edge and remote cloud to cope with challenging AI applications. Although the advantage of ECC greatly benefits various mobile applications, data security remains an important open issue in this scenario, which has not been well studied. This paper focuses on the profit maximization (PM) problem for security-aware task offloading in an ECC environment, i.e., considering the tasks from MDs with different service demands, edge nodes should decide them to be processed on the edge node or the remote cloud with a security guarantee. Specifically, we first construct the security model to measure the time overhead for each task under various scenarios. We then formulate the PM problem by jointly considering the security demand and deadline constraints of tasks. Finally, we propose a genetic algorithm-based PM (GA-PM) algorithm, the coding strategy of which considers the task execution location and execution order. Moreover, the crossover and mutation operations are implemented based on the coding strategy. Extensive simulation experiments with various parameters varying demonstrate that our GA-PM can achieve better performance than all the comparison algorithms. • The security model is built to measure the execution time of tasks under different parameters. • A genetic algorithm (GA)-based PM algorithm is proposed to implement task offloading and obtain optimal profit. • A coding strategy is devised by considering the tasks execution location and execution order. • Extensive simulation results demonstrate that GA-PM algorithm achieves the optimal profit. [ABSTRACT FROM AUTHOR]