Mutual authentication between the user and the cloud is an essential requirement for the user to access the public cloud in cloud computing. In order to solve the problems of password-based identity authentication scheme in the cloud computing environment and to improve the efficiency of authentication in the environments of mass users, a fair equality-determination protocol was presented by developing an equality-determination protocol of constant complexity. The protocol proposed can compare the passwords between the user and the cloud fairly and secretly. Based on the protocol, a two-party privacy-protected identity authentication scheme based on semantically secure algorithm-Bresson algorithm was further presented. Finally, the correctness of the scheme was proved and its security was analyzed.