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The telecare medicine information systems (TMISs) not only help patients to receive incessant health care services but also assist the medical staffs to access patients' electronic health records anytime and from anywhere via Internet. Since the online communications are exposed to numerous security threats, the mutual authentication and key agreement between patients and the medical servers are of prime significance. During the recent years, various user authentication schemes have been suggested for the TMISs.Nonetheless, most of them are susceptible to some known attacks or have high computational cost. Newly, an effective remote user authentication and session key agreement protocol has been introduced by Ravanbakhsh and Nazari for health care systems. Besides the nice contributions of their work, we found that it has two security weaknesses, namely, known session-specific temporary information attack and lack of perfect forward secrecy. As a result, to overcome these deficiencies, this paper suggests a novel anonymous and unlinkable user authentication and key agreement scheme for TMISs using the elliptic curve cryptosystem (ECC). We have evaluated the security of the proposed scheme by applying the automated validation of internet security protocols and applications (AVISPA) tool with the intention of indicating that our scheme can satisfy the vital security features. In addition, we have compared the proposed protocol with related schemes to show that it has a proper level of performance. The obtained results demonstrate that the new scheme is more preferable considering both efficiency and security criteria.