Internet of Things (IoT) is a new paradigm that has been evolving into the wireless sensor networks to expand the scope of networked devices (or things). This evolution drives communication engineers to design secure and reliable communication at a low cost for many network applications such as radio frequency identification (RFID). In the RFID system, servers, readers, and tags communicate wirelessly. Therefore, mutual authentication is necessary to ensure secure communication. Normally, a central server supports the authentication of readers and tags by distributing and managing the credentials. Recent lightweight RFID authentication protocols have been proposed to satisfy the security features of RFID networks. Using a serverless RFID system is an alternative solution to using a central server. In this model, both the reader and the tag perform mutual authentication without the need for the central server. However, many security challenges arise from implementing lightweight authentication protocols in serverless RFID systems. We propose a new secure serverless RFID authentication protocol based on the famous elliptic curve cryptography (ECC). The protocol also maintains the confidentiality and privacy of the messages, tag information, and location. Although most of the current serverless protocols assume secure channels in the setup phase, we assume an insecure environment during the setup phase between the servers, readers, and tags. We ensure that the credentials can be renewed by any checkpoint server in the mobile RFID network. Thus, we implement ECC in the setup phase (renewal phase), to transmit and store the communication credentials of the server to multiple readers so that the tags can perform the mutual authentication successfully while far from the server. The proposed protocol is compared with other serverless frameworks proposed in the literature in terms of computation cost and attacks resistance.