Vehicular Ad hoc Networks (VANETs) are a subset of the Internet of Things (IoT) that are used in smart traffic applications. Due to their high speed, mobility, and exposure to the environment, the security requirements for VANETs result in the conflicting design goals of protecting member privacy while also ensuring non-repudiation. Group signature schemes can fulfill these requirements, but often at the cost of expensive bilinear pairing operations. Furthermore, the cost of updating the group key information can be costly. Accordingly, this paper has two goals. First, it presents a group signature scheme that has been modified to remove pairing operations by caching computed values, while still preserving the critical requirement of conditional privacy. Second, this paper presents an argument for the abandonment of perfect forward and backward secrecy in VANET schemes in order to prevent the generation of keys that are never used, or used only once, and reduce the twin burdens of excessive key recalculation and key redistribution on the system.