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In this paper, a novel efficient remote user authentication scheme using smart cards based on elliptic curve discrete logarithm problem (ECDLP) has been proposed. A remote user authentication scheme is a two-party protocol whereby an authentication server confirms the identity of a remote individual logging on to the server over an untrusted, unsecured network. The password based authentication schemes are commonly used for authenticating remote users. Many passwords based schemes both with and without smart card have been proposed; each scheme has its merits and demerits. Our proposed scheme does not require verifier table and allows the user to choose their passwords. The proposed scheme also withstands message replying attack.