Outsourcing to a cloud storage brings forth new challenges for the efficient utilization of computing resources as well as simultaneously maintaining privacy and security for the outsourced data. Data deduplication refers to a technique that eliminates redundant data on the storage and the network, and is considered to be one of the most-promising technologies that offers efficient resource utilization in the cloud computing. In terms of data security, however, deduplication obstructs applying encryption on the outsourced data and even causes a side channel through which information can be leaked. Achieving both efficient resource utilization and data security still remains open. This paper addresses this challenging issue and proposes a novel solution that enables data deduplication while also providing the required data security and privacy. We achieve this goal by constructing and utilizing equality predicate encryption schemes which allow to know only equivalence relations between encrypted data. We also utilize a hybrid approach for data deduplication to prevent information leakage due to the side channel. The performance and security analyses indicate that the proposed scheme is efficient to securely manage the outsourced data in the cloud computing.