As Big Data cloud storage servers are becoming popular the shortage of disk space in the cloud becomes a problem. Data deduplication is a method to control the explosion growth of data on the cloud and most of the storage providers are finding more secure and efficient methods for their sensitive data. Recently, an interesting technique called signcryption has been proposed, in which both the properties of signature (ownership) and encryption are simultaneously implemented, with better performance than the traditional signature-then-encryption approach. According to the deduplication, we introduce a new method for a group of users that can eliminate redundant encrypted data owned by different users. Furthermore, we generate the tag which will be the key component of big data management. We propose a new primitive group signcryption for deduplication called verifiable hash convergent group signcryption (VHCGS) by adding the properties of group signcryption and the verification facilities for the storage server (third party).