Wireless broadcast transmission technology enables vehicles to communicate with other nearby vehicles and with nearby fixed equipment. Vehicles and equipment within transmission range establish a self-organizing network called Vehicular Ad-hoc Network (VANET). The communication in VANETs is vulnerable to message manipulation attacks. Thus, mechanisms should be applied to ensure both the authenticity and integrity of the data broadcast. Any cryptographic technique employed for authentication requires the use of a cryptographic key, and mechanisms to restore the system quickly when either long-term and short-term cryptographic keying material are leaked or expired. Such mechanisms must be carefully designed to satisfy both perfect-forward-secrecy and security against known-key attacks. To achieve this, there should be no direct dependencies among keying material. Unfortunately, many existing proposals for authentication are not fully effective in VANETs, since many of them do not take a key-management mechanism into consideration or they fail to satisfy the requirements for secure key-update. In this paper, we first present a case study demonstrating that dependency among keying material is an exploitable vulnerability that violates perfect-forward-secrecy, and results in known-key attacks and message forgery attacks. Secondly, we propose a new cryptographic-key update protocol that consists of two sub-protocols: a long-term-key update protocol (for updating the long-term cryptographic keying material) and a short-term-key update protocol (for session-key establishment). Our scheme is accompanied by both security and efficiency analysis: we provide a formal security proof and demonstrate efficiency by conducting extensive performance analysis. This is compared with the security and efficiency of existing schemes in public literature.