In this paper, we analyse the security of two mutual authentication protocols that have been recently proposed by Gao et al. (IEEE Access, 7:8376–8384, 2019), a hash-based protocol and a Rabin public key based protocol. Our security analysis clearly shows important security pitfalls in these schemes. More precisely, in each protocol, we introduce efficient approaches to desynchronize the tag and the reader/server. The proposed attacks are almost deterministic and the complexity of each attack is a session for the hash-based and three sessions for Rabin public key based protocol. In addition, in the case of the hash-based protocol, we extend the proposed desynchronization attack to a traceability attack in which the adversary can trace any given tag based on the proposed attack with probability of almost one. In the case of Rabin public key based protocol, we extend the proposed desynchronization attack to a tag impersonation attack with the success probability of one. Besides, we propose an enhanced version of the Rabin public key based protocol to provide a secure authentication between the tag and the reader. We evaluate the security of the proposed protocol formally using the Scyther tool and also in Real-or-Random model.