## 1 Introduction  
The development of the Industrial Internet of Things (IIoT) has initiated an evolution in which resource allocation is optimized via network connectivity and real-time data exchange among interconnected systems and devices. Within the framework of Cyber Physical Systems (CPSs), this interconnectivity is vital for the effective administration of urban services and infrastructure [1, 2]. Terminal devices, including smart meters and sensors, are of utmost importance in this ecosystem as they enable communication via the Internet and streamline the exchange of data among diverse urban systems. In time-sensitive smart city applications, however, traditional centralized approaches, which frequently rely on cloud servers for data storage, extraction, and analysis, present obstacles such as single points of failure and latency issues. To surpass these challenges and guarantee the uninterrupted functioning of smart city infrastructure, a device-edge-cloud framework has been implemented. Edge servers are strategically placed in proximity to IIoT devices to offer storage and computational assistance. This configuration effectively mitigates latency and improves the dependability of critical urban functions. By utilizing this decentralized methodology, smart cities are able to optimize the capabilities of IIoT technologies while minimizing the drawbacks linked to centralized systems. As a result, urban efficiency, sustainability, and resilience are enhanced. As IIoT progresses towards device-edge-cloud topologies, ensuring security becomes a crucial problem due to both logical and physical changes in the system architecture [3; 4; 5]. In situations such as intelligent factories, where sensors, users, routers, and edge stations work together, it is essential to prioritize secure communication. Authentication is crucial for building confidence between entities that communicate inside complex ecosystems [6; 7; 8]. This paper introduces the Lightweight Industrial IoT Authentication (LI2A) scheme, specifically designed to address the unique security concerns commonly encountered in industrial settings employing IIoT technology. The proposed system framework contains a Trusted Authority (_TA_), Users (\(C_{i}\)), and Industrial IoT devices (\(IIoT_{j}\)), creating a resilient framework for secure communication and authentication. The approach includes crucial security elements such as mutual authentication, authenticated message integrity, key agreement, soundness, forward secrecy, robustness against various attacks, and minimum resource overhead. When considering security in the context of IIoT, the decision between using symmetric key or public key-based cryptography is of utmost importance. This study specifically examines symmetric key authentication, with a particular emphasis on session key agreement, despite the presence of various other authentication systems. The effectiveness of symmetric key authentication systems, as discussed in [9; 10; 11; 12; 13], is assessed in terms of their capability to establish mutual authentication with edge nodes, while also ensuring participant anonymity and forward secrecy. The LI2A technique stands out by specifically targeting the drawbacks of conventional symmetric key authentication schemes. Forward secrecy (FS) is implemented to guarantee that, in the event of a long-term secret key being compromised, it is not possible to retrieve earlier-session keys. This functionality improves the security of communication channels, particularly in IIoT contexts where devices have limited resources. This paper presents a thorough security assessment, showcasing the efficacy of LI2A in mitigating many possible threats like impersonation, man-in-the-middle attacks, replay attacks, eavesdropping, and modification attacks. This paper also assesses the performance of LI2A in terms of storage requirements, communication costs, and compute complexity, in addition to considering security aspects such as mutual authentication, message integrity, and forward secrecy.