5G network will serve billions of people worldwide in the near future and protecting human privacy from being violated is one of its most important goals. In this paper, we carefully studied the 5G authentication protocols (namely 5G AKA and EAP-AKA') and a location sniffing attack exploiting 5G authentication protocols vulnerability is found. The attack can be implemented by an attacker through inexpensive devices. To cover this vulnerability, a fix scheme based on the existing PKI mechanism of 5G is proposed to enhance the authentication protocols. The proposed scheme is successfully verified with formal methods and automatic verification tool TAMARIN. Finally, the communication overhead, computational cost and storage overhead of the scheme are analyzed. The results show that the security of the fixed authentication protocol is greatly improved by just adding a little calculation and communication overhead.