Passive radio-frequency identification (RFID) tag has been used in many applications. While the RFID market is expected to grow, concerns about security and privacy of the RFID tag should be overcome for the future use. To overcome these issues, privacy-preserving authentication protocols based on cryptographic algorithms have been designed. However, to the best of our knowledge, evaluation of the whole tag, which includes an antenna, an analog front end, and a digital processing block, that runs authentication protocols has not been studied. In this paper, we present an implementation and evaluation of a fully integrated passive UHF RFID tag that runs a privacy-preserving mutual authentication protocol based on a hash function. We design a single chip including the analog front end and the digital processing block. We select a lightweight hash function supporting 80-bit security strength and a standard hash function supporting 128-bit security strength. We show that when the lightweight hash function is used, the tag completes the protocol with a reader-tag distance of 10 cm. Similarly, when the standard hash function is used, the tag completes the protocol with the distance of 8.5 cm. We discuss the impact of the peak power consumption of the tag on the distance of the tag due to the hash function.