The integration of advanced communication technologies with modern vehicular systems has driven the evolution of vehicular ad-hoc networks (VANETs). These networks enable a seamless exchange of road safety information between vehicles and traffic management authorities via wireless links. However, the open nature of these communication channels introduces significant risks to the privacy and security of transmitted messages. To address these challenges, Liet al. (IEEE Trans. Inf. Forensics and Security, vol. 19, pp. 9629–9642, 2024) proposed a lattice-based authentication scheme designed for fog-assisted VANETs. This protocol utilizes lattice cryptography to ensure resilience against quantum attacks and employs fog computing to tackle scalability issues. Despite these advancements, a detailed analysis uncovers several vulnerabilities and inefficiencies in their design. This study identifies an anonymity disclosure attack on Liet al.’s scheme, compromising its privacy guarantees. In addition, redundancies in the signature generation process impose excessive computational burdens on resource-constrained vehicular devices. To address these shortcomings, this work introduces a lattice-based anonymous batch-verifiable authentication (LBABVA) scheme. Rigorous security analysis proves the scheme’s security in the random oracle model, while efficiency evaluations reveal significant improvements. The proposed scheme reduces the computational cost of the signing phase to 14.57% and the signature verification phase to 83.99% of the corresponding costs of the previous design, highlighting its superior performance and suitability for practical applications.