## 2 Background and Related Work
### Related Work  
The characteristics of fog computing, as specified in Section 2.1, make existing cloud computing authentication protocols inadequate to apply to fog computing [34]. [PERSON] et al. [35] address securing communication and privacy in fog-based vehicular ad hoc networks (VANETs). They propose an approach for secure and lightweight conditional privacy-preserving authentication. The paper aims to establish secure communication while preserving privacy on VANETs as traditional authentication methods are inefficient and may expose sensitive information, such as vehicle ID, location data, movement path, and the scope of everyday routines. The proposed approach assumes fog-based VANETs, where vehicles and fog nodes communicate. The proposed mechanism employs conditional privacy, where vehicles disclose only necessary information for authentication while preserving anonymity. Cryptographic techniques like bilinear pairings and elliptic curve cryptography (ECC) are used for secure and efficient authentication, minimizing computation and communication overhead. Their results demonstrate a balance between security and efficiency in VANETs. The approach is suitable for resource-constrained vehicular environments. However, scalability aspects and the impact of dynamic network conditions are not addressed. In [36], [PERSON] et al. proposed a mutual authentication based on elliptic curve cryptography (ECC) and one-way hash functions. These functions focus on the challenge of establishing mutual authentication between fog nodes and end-users in fog computing environments. The proposed approach establishes secure mutual authentication by leveraging these lightweight cryptographic techniques while minimizing computational and communication overheads. The scheme undergoes a security analysis, demonstrating protection against known attacks. Validation using the security protocol animator (SPAN) of the automated validation of Internet security protocols and applications (AVISPA) tool confirms its effectiveness. However, this protocol requires the edge user to store an extra identity (ID), and no authentication is presented between the fog and the cloud. The protocol used in [37] is a lightweight authentication protocol designed for resource-constrained devices in the Industrial Internet of Things (IIoT). The IIoT ecosystem faces numerous security vulnerabilities, including industrial espionage and sabotage. The work designed an authentication protocol to operate within the IIoT environment. The environment has sensors, networks, and services that connect and control production systems. It assumes the presence of resource-constrained devices, which may have computational power, memory, and energy efficiency limitations. The authentication protocol uses lightweight  
Figure 1: Three-tier architecture. operations, including xor, addition, subtraction, and a hash function. The protocol minimizes communication overhead by facilitating authentication with four message exchanges between the participating entities. Furthermore, a performance and security comparison with state-of-the-art protocols reveals its performance efficiency for resource-constrained IIoT devices. The protocol achieves higher security levels comparable to computationally expensive schemes. Refs. [38; 39] raised mutual authentication in the Internet of Vehicles (IoV) environment. [PERSON] et al. [38] proposed an anonymous authentication scheme based on fog computing for vehicular ad hoc networks (VANETs). The problem lies in the privacy concerns of using real identities for vehicle authentication in VANETs. Existing privacy-protection schemes rely on anonymous authentication but face challenges related to network congestion and updating anonymous information, resulting in poor real-time performance and potential key leakage. The proposed scheme operates within the VANET environment, assuming the presence of vehicles, roadside units (RSUs), and a trusted authority (TA).