Communications between nodes in Vehicular Ad-Hoc Networks (VANETs) are inherently vulnerable to security attacks, which may mean disruption to the system. Therefore, the security and privacy issues in VANETs are entitled to be the most important. To address these issues, the existing Conditional Privacy-Preserving Authentication (CPPA) schemes based on either public key infrastructure, group signature, or identity have been proposed. However, an attacker could impersonate an authenticated node in these schemes for broadcasting fake messages. Besides, none of these schemes have satisfactorily addressed the performance efficiency related to signing and verifying safety traffic-related messages. For resisting impersonation attacks and achieving better performance efficiency, a Secure and Efficient Conditional Privacy-Preserving Authentication (SE-CPPA) scheme is proposed in this paper. The proposed SE-CPPA scheme is based on the cryptographic hash function and bilinear pair cryptography for the signing and verifying of messages. Through security analysis and comparison, the proposed SE-CPPA scheme can accomplish security goals in terms of formal and informal analysis. More precisely, to resist impersonation attacks, the true identity of the vehicle stored in the tamper-proof device (TPD) is frequently updated, having a short period of validity. Since the MapToPoint hash function and a large number of cryptography operations are not employed, simulation results show that the proposed SE-CPPA scheme outperforms the existing schemes in terms of computation and communication costs. Finally, the proposed SE-CPPA scheme reduces the computation costs of signing the message and verifying the message by 99.95% and 35.93%, respectively. Meanwhile, the proposed SE-CPPA scheme reduces the communication costs of the message size by 27.3%.