Authentication of communicating entities and confidentiality of transmitted data are fundamental procedures to establish secure communications over public insecure networks. Recently, many researchers proposed a variety of authentication schemes to confirm legitimate users. Among the authentication schemes, a one-time password authentication scheme requires less computation and considers the limitations of mobile devices. The purpose of a one-time password authentication is to make it more difficult to gain unauthorized access to restricted resources.This paper discusses the security of Kuo-Lee's one-time password authentication scheme. Kuo-Lee proposed to solve the security problem based on Tsuji-Shimizu's one-time password authentication scheme. It was claimed that their proposed scheme could withstand a replay attack, a theft attack and a modification attack. Therefore, the attacker cannot successfully impersonate the user to log into the system. However, contrary to the claim, Kuo-Lee's scheme does not achieve its main security goal to authenticate communicating entities. We show that Kuo-Lee's scheme is still insecure under a modification attack, a replay attack and an impersonation attack, in which any attacker can violate the authentication goal of the scheme without intercepting any transmitted message. We also propose a scheme that resolves the security flaws found in Kuo-Lee's scheme.