With the rapid development of cloud computing and the arrival of large data age, users are confronted with huge data and information to be processed which means massive amounts of difficult tasks. Consequently, how to securely outsource some time-consuming computing tasks to an untrusted public cloud server has aroused widespread concern. To realize the data privacy protection and the verifiability of calculation results in outsourcing computing, based on the single server model, this paper proposes a new privacy-preserving protocol for outsourcing power exponent on a group field, called GEXP(outsourcing power exponent on a group field). The scheme can prevent adversaries from getting any input/output data. Moreover, it effectively avoids the collusion attack in the dual server model. Compared with the existing schemes, GEXP can detect the wrong result returned by the cloud server with 100% probability, which ensures that the user can fully verify the result of outsourcing calculation. The formal security analysis and experiments indicate that our scheme is to protect privacy and highly efficient. In experiments, we compare our scheme with other state-of-the-art schemes to further demonstrate the superiorities in security and efficiency. In addition, in order to prove the practicality of our scheme, this paper gives the specific application of GEXP in cloud storage data integrity verification.