Vehicular ad-hoc networks (VANETs) can substantially improve traffic safety and efficiency by providing a communication platform between vehicles and roadside units (RSUs) to share real-time information on traffic and road conditions. Two essential security requirements for VANETS are data authentication and the preservation of the privacy of vehicle owners. Conditional privacy-preserving authentication (CPPA) schemes address both of these security requirements. The existing CPPA schemes either require a tamperresistant device (TRD), which is vulnerable to key exposure based on physical attacks, or require continuous communications of vehicles with RSUs, which significantly increases the communication overhead. This paper addresses both of these problems by proposing a provable secure, and efficient CPPA scheme. We prove the privacy-preserving property of our scheme in the random oracle model and show that it offers anonymity, unlinkability, and tamper detection even if a physical attacker succeeds in compromising an individual OBU. Moreover, the performance analysis of our scheme shows a substantial improvement in communication cost, especially in comparison with RSU-aided schemes that require continuous vehicle communication with roadside units and a Trusted Authority (TA).