Recently, Vehicular Ad hoc Networks (VANETs) have gained extensive attention in both academia and industry. VANETs play a seminal role in smart transportation by enhancing driving convenience and traffic efficiency through real-time information interaction. Despite this, ensuring secure authentication and privacy preservation remains two challenging issues in VANETs. The existence of security schemes that are vulnerable to privacy and security issues or have substantial computation and communication overheads shows the necessity for further research in this area. Therefore, in this article, we present an anonymous authentication scheme based on Zero-Knowledge Proof. To be more specific, we adopted Simulation Extractable Zero-Knowledge SNARKs (SE zk-SNARKs) to achieve anonymity and conditional privacy. Although zk-SNARK proofs are succinct and fast to verify, the time required for proof generation poses a major obstacle in using zk-SNARKs in a time-sensitive VANET environment. To overcome this drawback, we focus on key components of the proof statement, separating the repetitive calculations, and archiving in the Blockchain, which serves as an immutable data storage. By omitting repetitive calculations, our proposed scheme becomes lightweight enough to run efficiently on VANETs. Security analysis showed that the proposed protocol satisfied security and privacy requirements. Additionally, a computation cost analysis is provided to demonstrate the significant advantage of our proposed scheme. ProVerif is used to verify the strong secrecy of the protocol, and the results show that privacy can be guaranteed in the proposed scheme.