Cloud computing provides a new paradigm of computing. It offers a scalable, manageable and huge pool of resources that can be accessed by users from anywhere anytime. It also ensures the integrity of data stored on the cloud. But ensuring the confidentiality and integrity of sensitive information is still a big challenge. To overcome this challenge, a hybrid two-phase security system for preserving the privacy of data on the cloud has been proposed. The hybrid approach combines feature extraction and encryption techniques to enhance the security of accessing data from the cloud. At first, the minutiae point has been extracted from the biometric fingerprint, locally collected from the state university in Northern India. The private key has been finalized by generating an elliptic curve using the minutiae point for achieving better encryption of fingerprint. The effectiveness of the approach has been tested in terms of similarity score, False Matching Ratio (FMR), False Non Matching Ratio (FNMR) and recognition accuracy, when applied on the local fingerprint database. The evidence of the outcomes suggests that the proposed technique ensures relatively improved security and privacy of data in the cloud system as compared to some recent state-of-art methods.