Security and privacy of vehicles, occupants of such vehicles, roadside infrastructure, and other entities that are part of Vehicular Ad hoc NETwork (VANET) cannot be overstated. Cryptography is commonly used to authenticate and to secure communication among VANET entities. As vehicles are mobile, it is essential for authentication protocols to be lightweight, quick, and with minimal number of passed messages. It is also necessary to ensure that these protocols are secure against attacks. However, extant authentication protocols are not necessarily lightweight and almost all of them are vulnerable to relay attacks. We propose secure and truly lightweight authentication protocols for the VANET environment.