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Radio Frequency IDentification based systems are getting pervasively deployed in many real-life applications in various settings for identification and authentication of remote objects. However, the messages that are transmitted over a insecure channel, are vulnerable to security and privacy concerns such as data privacy, location privacy of tag owner and etc. Recently, Yeh et al.'s proposed a RFID authentication protocol based on quadratic residue which is claimed to provide location privacy and prevent possible attacks. In this paper, we formally analyzed the protocol and we proved that the protocol provides destructive privacy according to Vaudenay privacy model. Moreover, we proposed a unilateral authentication protocol and we prove that our protocol satisfies higher privacy level such as narrow strong privacy. Besides, we proposed an enhanced version of our proposed protocol, which has same privacy level as Yeh at al protocol, but has reader authentication against stronger adversaries. Furthermore, the enhanced version of our protocol uses smaller number of cryptographic operations when compared to Yeh at al protocol and it is also cost efficient at the server and tag side and requires O(1) complexity to identify a RFID tag.