## 1 Introduction
Comparisons with other schemes emphasize the efficiency and appropriateness of LI2A for use in IIoT devices that have limited resources and are frequently used in industrial environments. LI2A presents itself as a highly promising option for ensuring the security of communications in Industrial IoT settings. It provides a lightweight yet strong authentication architecture. This scheme not only fulfills rigorous security criteria but also demonstrates exceptional performance, rendering it very suitable for the dynamic field of industrial automation. The utilization of LI2A has the capacity to improve the security of industrial systems, hence facilitating the dependable and secure incorporation of IIoT devices into critical processes. Below is a summary of this paper's contributions:  
1. The LI2A scheme contributes a tailored authentication method designed specifically for Industrial Internet of Things (IIoT) environments. It addresses the unique challenges presented by IIoT devices in industrial settings, ensuring a robust and secure communication framework. 2. LI2A introduces a comprehensive set of security features, including mutual authentication, message integrity, key agreement, soundness, forward secrecy, and resilience against diverse attacks. This ensures a high level of security, fostering mutual trust between IIoT devices and users while preventing various types of potential threats. 3. The scheme demonstrates efficiency in resource utilization, particularly with minimal storage requirements and communication overhead. This is crucial for IIoT devices with limited computing and storage capacities, making LI2A well suited for deployment in resource-constrained industrial settings. 4. LI2A's performance evaluation showcases its superiority in terms of communication and computation costs compared to alternative schemes. The scheme's lightweight design does not compromise on security, making it a practical and efficient solution for securing communications in dynamic industrial automation environments. 5. The performance evaluation of LI2A highlights its superiority in terms of communication and computation costs compared to alternative schemes. This feature enhances the confidentiality of communication and adds an additional layer of protection against potential breaches. Related WorkRef. [14] introduces an authentication scheme for IoT-based RFID systems, employing symmetric cryptography. The authors' approach involves a shared key between the server and tag device for mutual authentication, with key updates occurring in each session. However, concerns arise regarding the synchronicity of the shared key state. Ref. [15] devised a symmetric key-based authentication scheme for servers and sensors. In their storage-less authentication proposal, the shared key remains unchanged. Notably, auxiliary information accompanies the shared key to authenticate identities; although this information is updated each session, it is transmitted over a public channel. Consequently, if the shared key is compromised, all previous sessions become insecure. The authors of [16] extended their work with a symmetric AKA authentication scheme that ensures unlinkability. In this scenario, user equipment (UE) and home network (HN) share a pre-established key for mutual authentication. The scheme protects previous sessions by preventing the calculation of the session key, even if the pre-shared key is exposed. Ref. [17] presents an improved and lightweight authentication scheme based on [18]. The user and gateway, as well as the gateway and IoT device, share a key for message integrity protection. Importantly, the shared key remains static throughout the entire session, posing a risk where a compromised IoT device or user exposes the shared key, granting access to all messages, including the session key. Ref. [19] proposed a scheme where keys are shared between adjacent layers in the authentication process, mirroring the vulnerabilities of [17] in terms of compromised shared keys jeopardizing the security of the communication channel. Ref.