In WiSec'08, Song and Mitchell proposed an RFID mutual authentication protocol. Song also extended this protocol for RFID tag ownership transfer. These two protocols are designed to have the most security properties in the literature. We discover that, however, the mutual authentication protocol is vulnerable to both tag impersonation attack and reader impersonation attack, which enable an adversary to impersonate any legitimate reader or tag. We also discover that the ownership transfer protocol is vulnerable to a de-synchronization attack, which prevents a legitimate reader from authenticating a legitimate tag, and vice versa. We analyze the vulnerabilities of these protocols and propose our revisions to eliminate the vulnerabilities with comparable storage and computational requirements.