As anonymity increasingly becomes a necessary and legitimate aim in many applications, a number of anonymous authentication schemes have been suggested over the years. Among the many schemes is Lee and Kwon's password-based authentication scheme for wireless environments. Compared with previous schemes, Lee and Kwon's scheme not only improves anonymity by employing random temporary IDs but also provides user-friendliness by allowing human-memorable passwords. In this letter, we point out that Lee and Kwon's scheme, despite its many merits, is vulnerable to off-line password guessing attacks and a forgery attack. In addition, we show how to eliminate these vulnerabilities.