Vehicular ad hoc network is a multihop mobile wireless communication network that can realize many vehicle-related applications through multitop communication. In the open wireless communication environment, security and privacy protection are important contents of VANET research. The most basic method of VANET privacy protection is anonymous authentication. Even through, there are many existing schemes to provide anonymous authentication for VANETs. Many existing schemes suffer from high computational cost by using bilinear pairing operation or need the assistance of the trust authorities (...) during the authentication process or rely on an ideal tamper-proof device, which requires very strong security assumption. In this study, an anonymous authentication and key negotiation scheme by using private key and group key is proposed, which is based on pseudonym using the nonsingular elliptic curve. In this scheme, there is no third party trust center to participate in the authentication, there is no need to query the database, and there is no need of the local database to save the identity information of many vehicles, which reduce the storage space and the authentication time compared with other schemes. The proposed scheme only needs realistic TPDs. In the proposed scheme, TPDs do not need to preinstall the system key as many other schemes do; hence, the failure of a single TPD does not affect the security of the entire system. The security of the scheme is proved under the random oracle model. Compared with the related schemes using bilinear pairings, the computational cost and communication cost of the proposed scheme are reduced by 82% and 50%, respectively. (shrink)