User authentication technologies that provide the anonymity of users have been proposed according to increase in the need for security of personal information and privacy protection in ubiquitous computing environments. Studies on smart card-based authentication systems have been actively conducted. In 1993, Chang and Wu first proposed a smart card-based remote user authentication system. In 2005, Das et al. proposed a remote user authentication scheme using bilinear pairings. In the Das et al.'s scheme, it prevents many logged in users with the same login-ID and provides a flexible password change option to the registered users without any help of the remote system. In this paper, it shows the weakness of the Das et al.'s scheme for an impersonation attack using the Euclidean algorithm in the login process. This paper proposes an improved scheme that shows more safe and effective figures than the scheme proposed by Das et al.'s scheme.