This study addresses the critical challenge of achieving reliable message exchange among vehicles in Vehicular Ad-hoc Networks (VANETs). It is crucial to quickly share safety messages, traffic updates, available services, and road conditions among vehicles in VANETs. For security reasons, messages must originate exclusively from authenticated vehicles, ensuring secure message exchange and data privacy. Numerous schemes for privacy-preserving authentication have been proposed, yet they suffer from constant service provider involvement and the requirement for vehicles to generate parameters on the fly. To address these challenges, this paper introduces LbPV, a secure lattice-based privacy-preserving mutual authentication scheme emphasizing the reliability of messages, preventing spoofing and unauthorized access. LbPV eliminates service provider monitoring and allows vehicles to authenticate messages without generating parameters while on the move. This is achieved by exchanging a confidential token using lattice-based encryption and message signing. By verifying the signed message and using the shared token, receiving vehicles can confirm the authenticity of the messages. By using lattice-based cryptography, the proposed protocol is also designed with the potential to resist future attacks, including quantum attacks, enhancing its long-term security viability. The security analysis of LbPV includes formal and informal evaluations that demonstrate its robustness. Performance evaluations using the NTL library show that LbPV, with service provider parameters, outperforms existing approaches in the literature. Results of performance analysis indicate that when compared to the most efficient traditional non-lattice-based scheme discussed in this paper, the proposed protocol has increased computation cost, communication cost, and power consumption by 77.71%, 98.58%, and 77.71%, respectively. Conversely, when compared to the most efficient lattice-based scheme discussed in this paper, the proposed scheme demonstrates reductions in computation cost, communication cost, and power consumption by 71.95%, 2.16%, and 71.95%, respectively.