Given the ever-increasing advances in science and technology in recent years, the security and authentication issues using elliptic curve cryptography (ECC) have gained a lot of attentions especially for smart cards in a variety of networks, such as smart homes and medical-based care systems being based on the Internet of things (IoT). In systems such as health care, patient information is always crucial and no one should have access to this information. Recently, much research has been conducted in the area of security and authentication for medical care systems being based on the IoT using RFID technology. One of these schemes is the Alamr et al.’s protocol, which bears high computational cost for authentication between card and reader. In this paper, we have proposed a scheme based on ECC in order to establish a mutual authentication within RFID technology in the IoT. In the proposed scheme, computational cost, communication cost and elliptic curve point multiplication running time and data storage cost are investigated. Moreover, security requirements and various attacks have also been considered in the proposed protocols. AVISPA software has also been used for security analysis. The analyses represent that the proposed scheme has lower computational costs, lower communication costs and less elliptic curve point multiplication time compared with those of similar protocols. It has also resolved the security shortcomings of the RFID authentication protocol.