Password-based remote user authentication is a hotspot in authentication protocol research. The security of a proposed remote user authentication scheme, the Lee-Kim-Yoo protocol, advanced by Sung-Woon Lee is analyzed, which used nonce random and had very low computational costs. However, this scheme still has many security faults. The weakness of the scheme is demonstrated. This paper proposes an improved scheme, a novel nonce and hash-based remote user authentication scheme using smart cards and analyses the amended protocol with BAN logic. It indicates that the amended protocol can reach the goal of the protocol.