Privacy preservation in RFID systems is a very important issue in modern day world. Privacy activists have been worried about the invasion of user privacy while using various RFID systems and services. Hence, significant efforts have been made to design RFID systems that preserve users' privacy. Majority of the privacy preserving protocols for RFID systems require the reader to search all tags in the system in order to identify a single RFID tag which not efficient for large scale 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 is to perform group-based authentication that improves the tradeoff between scalability and privacy by dividing the tags into a number of groups. This novel authentication scheme ensures privacy of the tags. However, the level of privacy provided by the scheme decreases as more and more tags are compromised. To address this issue, in this paper, we propose a group based anonymous private authentication protocol (AnonPri) that provides higher level of privacy than the above mentioned group based scheme and achieves better efficiency (in terms of providing privacy) than the approaches that prompt the reader to perform an exhaustive search. Our protocol guarantees that the adversary cannot link the tag responses even if she can learn the identifier of the tags. Our evaluation results demonstrates that the level of privacy provided by AnonPri is higher than that of the group based authentication technique.