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Smart home systems can provide health care services for people with special needs in their own homes. Briefly defined, such a smart home has special electronics to enable the remote control of automated devices specifically designed for remote health care to ensure the safety of the patient at home and the supervision of their health status. These sensors are linked to a local intelligence unit responsible for analyzing sensor data, detecting emergency situations, and interfacing between the patient at home and a set of people involved in their health care, such as doctors, nurses, emergency services, and paramedics. Smart homes can improve the patient's quality of life and safety through the innovative use of advanced technologies. Telemedicine and telecare are driving forces behind the adoption of smart homes. The telecare medicine information system (TMIS) has drawn worldwide attention for the past 20 years, as modern technologies have made remote delivery of healthcare a reality. TMIS using multidisciplinary research and application involves advanced technologies in information processing, telecommunications, bio-sensing, and artificial intelligence including smart technologies. TMIS leverages the latest mobile and wireless communication technologies and widely available internet infrastructure to deliver quality services to home patients enabling them to remotely access information about their health and obtain telemedical services. TMIS delivers capabilities to remotely provide $24\times 7$ health care facilities to patients. Its purpose is to provide patients with convenient and expedited remote health care services, greatly improving the quality and efficiency of health care services. However, the open and insecure nature of the internet poses a number of security threats to patient secrecy and privacy. Security design for TMIS is not trivial. Essential security and privacy are provided by mutual authentication and key agreement protocols. This paper proposes an efficient and secure, bilinear pairing-based, unlink-able, mutual authentication and key agreement protocol for TMIS. The proposed protocol adopts a fuzzy extractor for the identification of patients using the biometric data. The security of the proposed protocol is based on the hardness of the elliptic curve discrete logarithm problem (ECDLP) and elliptic curve computational Diffie-Hellman problem (ECCDHP) to preserve the privacy of the user. The detailed security analysis is discussed, and the results of comparison are provided.