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A lightweight radio frequency identification (RFID) tag-reader mutual authentication scheme was recently proposed as an improvement over the original authentication protocol specified under the EPC Class 1 Generation 2 UHF RFID Protocol Standard (otherwise known as the "EPC Gen2" standard in short). The improved scheme seeks to protect the access password of the RFID tag against exposure to adversaries. In this paper, we show the weakness in this scheme by launching an attack that effectively exposes the access password. Thereafter, we propose some possible fixes to the scheme to protect it against the attack. We also present some experiment results, which show that the fixed schemes provide greater resistance against exposure of the access password. Based on insights gained from this work, we find that designing a secure authentication scheme by relying only on the minimal features available on an RFID tag is an extremely challenging task. Furthermore, we also stress on the need to put a proposed scheme through stringent tests to ascertain its effectiveness and resistance against attacks.