Vehicular Ad-hoc Networks (VANETs) are with great potentials to facilitate traffic management and improve driver safety. Blockchain-based conditional privacy-preserving authentication (BCPPA) is proposed to achieve an optimal tradeoff among anonymity, traceability and key/certificate management in VANETs. Existing BCPPA protocols mitigate these security and privacy challenges by adding a significant cost on verification and traceability. As a result, current solutions fail to meet high mobility, low latency, and real-time requirements of VANETs. In this article, we design three new system building blocks namely key derivation (KeyDer), signatures of knowledge (SoK) and smart contract, following by a more efficient BCPPA protocol (named as EBCPA). To show the advantage of EBCPA, we first demonstrate it can satisfy the necessary requirements (e.g., message authentication, conditional privacy protection, resilience to common attacks, and so forth). Moreover, we implement the EBCPA in the on-line Ethereum test network (Rinkeby), Hyperledger test network and VANETs simulation environment (via VanetMobiSim and NS-2). Finally, we evaluate its communication overhead and computational cost via comparing to existing BCPPA protocols that strive to achieve similar properties. From the implementation and comparison results, our proposal can improve efficiency by reducing the time cost of traceability at least 48.95% and verification at least 42.21%.