To reduce the computational load on both the back-end database and the tags, Ha et al. proposed a low-cost and strong-security (LCSS) RFID authentication protocol. In this paper, we identify two effective attacks, a desynchronization attack and a spoofing attack, against the LCSS protocol. The former attack can break the synchronization between the RFID reader and the tag in a single protocol run so that they can not authenticate each other in any following protocol run. The latter can impersonate a legal tag to spoof the RFID reader by extracting the ID of a specific tag during the authentication process. Moreover, we point out the potential countermeasure by adding an integrity check to improve the security.