Cloud computing with other emerging information technologies promotes the transformation and upgrading of the service industry. The new mode of cloud service brings the convenience and agility by the remote service and on-demand use. Meanwhile, it also expands the existing information security boundary, trigging new security problems. Trust mechanism provides a good solution for the security problem of cloud service. This paper builds a service-oriented architecture of task assigning system in the cloud environment. It introduces the trust mechanism into cloud services by measuring the trust requirement of tasks and the trust degree of service resources. Considering the execution time, cost and trust as optimization objectives, we propose a business process driven multi-objective task assignment model in the cloud service system. Some typical structures: sequence, parallel, parallel-AND, parallel-OR, parallel-XOR, simple loop and combined loop are introduced to represent the functions of business process structures. The three objectives are set to keep security and trust of the system based on high efficiency and low cost. An improved strength Pareto genetic algorithm 2 (SPGA2) with the local search strategy is proposed to improve the search efficiency of solution space in multi-objective task assignment problem. Finally, simulation experiments verify the availability of the model and the superiority of the algorithm.