In a Vehicular Ad Hoc Network (VANET), numerous vehicles are interconnected through a wireless network to facilitate communication. The primary objective of a VANET is to enhance driver safety and comfort by enabling the exchange of traffic-related messages within the vehicular environment. These messages can include vital information such as traffic conditions, accident alerts, and road hazards. However, addressing the security challenges in VANETs is paramount to avoid serious vulnerabilities that can compromise the entire network. One of the critical security challenges is conditional privacy-preserving authentication. This requirement mandates that each vehicle must be authenticated by other vehicles or Roadside Units (RSUs) while ensuring the privacy of the vehicle's identity. Moreover, it is essential to have the capability to trace a malicious user under specific conditions, such as in the event of a security breach or misuse of the network. In this research, we conduct an in-depth cryptanalysis of a recently proposed aggregate signature scheme designed for authentication in VANETs with conditional privacy-preserving property. Our analysis identifies the existing scheme is vulnerable against a malicious Key Generation Center (KGC) attacker, in contrast to the authors' claims. To address these issues, we propose a novel, secure, and efficient authentication scheme that maintains the conditional privacy-preserving property. We evaluate our scheme and provide a formal security proof within the Random Oracle Model (ROM). In addition to enhancing security, our scheme improves efficiency by reducing the computational and communication overhead typically associated with authentication processes in VANETs. This makes our solution not only secure but also practical for real-world deployment.