In this modern era of smart cities supported by digital locker facilities, e-governance, e-healthcare, e-banking, e-learning systems, and other digital systems, People use mobile phones to upload and share sensitive documents with others. Previous group key management schemes suffer from problems such as expensive double encryptions, the need for re-encryption during every user leave operation and becoming vulnerable to attacks. Hence, to address these three issues, this manuscript introduces a pairing based cryptography based computationally efficient protocol powered by polynomial equations. First, to reduce computational overhead during group communication, the document is encrypted only once with minimal complexity before uploaded to cloud.Second, during a user leave or join operation, the re-encryption of documents in cloud is avoided in the group manager side by tactically relieving the group key parameters from the polynomial parameters. Finally, the security analysis shows that the careful handling of secure parameters of users averts the collusion attack, man-in-the-middle attack, message modification attacks, and all other possible attacks. The implementation results portray the capacity of this work to handle secure group communications in cloud storage.