For meeting the demands of safety, traffic management, and high mobility, vehicular adhoc network (VANET) has become a promising component for smart transportation systems. However, the wireless environment of vehicular network leads to various challenges in the communication security. Hence, several authentication schemes have previously been proposed to address VANET security issues but their procedures disregard the balance between effectiveness and security. Thus, this paper presents a new decentralized authentication protocol that relies on lightweight functions such as the Chebyshev chaotic map and logical shift operator to achieve the high mobility requirement. In order to reduce the number of messages transferred over the network, this protocol attempts to eliminate any redundant authentication steps during its authentication stage. Additionally, the new protocol solves key management problems by applying a little modification to the public key infrastructure to ignore certificates transmission over the network. The proposed design incorporates the self-authentication concept to safeguard the vehicle trip route on the road. Moreover, the performance evaluation is conducted to verify that the proposed protocol outperforms the most related scheme in terms of security and efficiency aspects. Finally, the Scyther simulation validates the security robustness of the new protocol.