ABSTRACT: In heterogeneous wireless networks, the industrial Internet of Things (IIoT) is an essential contributor to increasing productivity and effectiveness. However, in various domains, such as industrial wireless scenarios, small cell domains, and vehicular ad hoc networks, an efficient and stable authentication algorithm is required (VANET). Specifically, IoT vehicles deal with vast amounts of data transmitted between VANET entities in different domains in such a large-scale environment. Also, crossing from one territory to another may have the connectivity services down for a while, leading to service interruption because it is pervasive in remote areas and places with multipath obstructions. Hence, it is vulnerable to specific attacks (e.g., replay attacks, modification attacks, man-in-the-middle attacks, and insider attacks), making the system inefficient. Also, high processing data increases the computation and communication cost, leading to an increased workload in the system. Thus, to solve the above issues, we propose an online/offline lightweight authentication scheme for the VANET cross-domain system in IIoT to improve the security and efficiency of the VANET. The proposed scheme utilizes an efficient AES-RSA algorithm to achieve integrity and confidentiality of the message. The offline joining is added to avoid remote network intrusions and the risk of network service interruptions. The proposed work includes two different significant goals to achieve first, then secure message on which the data is transmitted and efficiency in a cryptographic manner. The Burrows Abdi Needham (BAN logic) logic is used to prove that this scheme is mutually authenticated. The system's security has been tested using the well-known AVISPA tool to evaluate and verify its security formally. The results show that the proposed scheme outperforms the ID-CPPA, AAAS, and HCDA schemes by 53%, 55%, and 47% respectively in terms of computation cost, and 65%, 83%, and 40% respectively in terms of communication cost.