Privacy protection is a very important issue during authentications in RFID systems. In order to achieve high-speed authentication in large-scale RFID systems, researchers propose tree-based approaches, in which any pair of tags share a number of key components. Another technique can be to perform group based private authentication that improves the tradeoff between scalability and privacy by dividing the tags into a number of groups. This is a novel authentication scheme that ensures privacy of the provers. However, one limitation of this technique is that the level of privacy provided by the scheme decreases as more and more tags are compromised Therefore, in this paper, we propose a group based anonymous private authentication protocol (AnonPri) that provides higher level of privacy than the above mention group based scheme and achieves better efficiency than the approaches that prompt the reader to perform an exhaustive search. Our protocol provides unlinkability and thereby preserves privacy. The adversary cannot link the responses with the tags, even if she can learn the identifier that the tags are using to produce the response. To evaluate AnonPri, we have compared both the protocols, AnonPri and the group based authentication. The experiment results establish that the level of privacy provided by AnonPri is higher than that of the group based authentication.