Vehicle ad hoc networks (VANETs) is a promising network scenario for greatly improving traffic efficiency and safety, in which smart vehicles can communicate with other vehicles or roadside units. For the availability of VANETs, it is very important to deal with the security and privacy problems for VANETs. In this paper, based on certificateless cryptography and elliptic curve cryptography, we present a certificateless signature with message recovery (CLS-MR), which we believe are of independent interest. Then, a practical certificateless conditional privacy preserving authentication (PCPA) scheme is proposed by incorporating the proposed CLS-MR scheme. Furthermore, the security analysis shows that PCPA satisfies all security and privacy requirements. The evaluation results indicate that PCPA achieves low computation and communication costs because there is no need to use the bilinear pairing and map-to-point hash operations. Moreover, extensive simulations show that PCPA is feasible and achieves prominent performances in terms of message delay and message loss ratio, and thus is more suitable for the deployment and adoption of VANETs.