To achieve efficient vehicular network communication and service, researchers proposed fog-based vehicular networks (FVNs). One of the prerequisites for developing largescale FVNs is ensuring the security and privacy of the entire network environment. However, the existing schemes proposed for FVNs exist considerable calculation and communication costs and/or security vulnerabilities. Therefore, to promote efficient FVN authentication, we propose a lightweight security protocol using self-certified public key cryptography. In the protocol, the trusted authority does not need to participate in the authentication process between the vehicle and the fog node online. And the vehicle can dynamically update its login password and pseudonym, without performing complicated interactive steps with the trusted authority. In addition, our protocol supports batch verification, which significantly improves the system authentication efficiency. A detailed security analysis reveals that our protocol can meet the security requirements of vehicular networks while resisting the common types of attack. Calculation and communication overhead comparisons further prove that our protocol exhibits better performance than related schemes.