In cloud computing, the combinations of various computing units are globally separated, however, electronically connected. In recent years, the number of cloud users has adequately increased in the modern societies. In order to restrict the illegal access of the intruders, a strong user authentication is required for cloud computing environment. In this regard, in this paper, we came up with a computationally productive and safer anonymous authentication pattern for cloud users. Moreover, in our scheme, mutual authentication is performed in an anonymous way between the cloud user and the cloud server to strongly verify the legitimacy of each user before providing an access into the cloud. After the successful anonymous mutual authentication, the targeted cloud service provider and the cloud users make mutual communication between them. In case of any misbehaviouring after successful mutual authentication, an efficient revocation mechanism is proposed through which the trusted third party (TTP) can revoke the cloud users or service providers from the cloud environment. Even though there were many anonymous cloud computing authentication schemes, the previously proposed authentication schemes suffered from high computational cost during certificate verification. Moreover, the section related to analysis proves that the explained scheme is robust enough in terms of security strength to avoid various security attacks and breeches.