## 4 Security Analysis  
1. Mutual Authentication: The process of mutual authentication embedded within the LI2A framework is a cornerstone of its security architecture. This intricate mechanism ensures that \(I1 IoT_{j}\) authenticates \(C_{i}\) and vice versa, fostering a mutual trust environment. \(IIoT_{j}\), during the authentication process, meticulously checks the received message \(m_{3}\), conducting computations such as \(ID_{i}=(m_{1}\oplus K_{s})\), \(ID_{j}=(m_{2}\oplus K_{s})\), and \(T_{i}=h(T_{m_{3}}\|K_{s})\). These computations generate unique identifiers and a hash value, forming the basis for the verification process, which \(I1 IoT_{j}\) undertakes to confirm the identity of \(C_{i}\). Reciprocally, \(C_{i}\) engages in the same authentication process, leveraging message \(m_{4}\) to compute \(T_{i}=h(T_{ACK}\|K_{s})\) and ensuring its congruence with the received \(T_{ACK}\). This mutual authentication not only establishes a secure identity confirmation but also solidifies the foundation of trust between \(I1 IoT\) entities. Thus, the LI2A scheme not only supports mutual authentication but elevates it to a critical security parameter in IIoT ecosystems, fostering a robust and trustworthy communication paradigm. 2. Message Integrity: A fundamental aspect of secure communication is the assurance of message integrity. The LI2A scheme addresses this with a focus on the secure exchange of information between entities. Shared session keys play a pivotal role in this process, acting as cryptographic anchors that detect any unauthorized alterations to the transmitted data. The LI2A scheme employs cryptographic mechanisms to detect and prevent unauthorized changes, ensuring that the communicated information remains untampered. In the event of any deviation, the system promptly identifies and flags a verification failure, providing a robust defense against potential security breaches. This commitment to preserving the unaltered nature of communication underscores the LI2A scheme's dedication to reliable and trustable data exchanges in IIoT environments. Figure 4: Forward secrecy phase. Session Key Agreement: The exclusive sharing of session keys (\(K_{s}\)) between Users (\(C_{i}\)) and IIoT devices (\(IIoT_{j}\)) is a pivotal aspect of the LI2A security framework. During the initial registration phase, \(IIoT_{j}\) dynamically generates the session key using \(K_{s}=h(\tau\|ID_{i}\|ID_{j}\|K_{m}\|r_{0})\)), subsequently confirming \(C_{i}\)'s identity. Simultaneously, \(C_{i}\) independently computes the session key \(K_{s}=h(\tau\|ID_{i}\|ID_{j}\|K_{m}\|r_{0}))\) and verifies its consistency by validating \(m_{4}=(Ack,T_{ACK},h(T_{ACK}\|K_{s}))\). This intricate yet secure process ensures a robust and verifiable agreement on the session key, enhancing the overall security posture of the LI2A scheme. The session key agreement mechanism not only provides a secure foundation for communication but also establishes a dynamic and adaptive security posture that is crucial for addressing evolving threats in IIoT ecosystems. 4. Soundness: A critical security parameter is exemplified in the LI2A scheme through a meticulously designed process. Upon receiving the message from the \(TA\), \(IIoT_{j}\) computes the session key (\(K_{s}\)) and seamlessly incorporates it into the validity value \(m_{3}\) for \(C_{i}\)'s authentication. Simultaneously, \(C_{i}\) engages in a parallel process, computing its session key (\(Ks\)) upon receiving the message from \(TA\) and utilizing a validation mechanism. This synchronous and coordinated approach guarantees the soundness of the LI2A scheme, emphasizing the accuracy and uniformity of cryptographic key usage. The assurance of soundness not only validates the legitimacy of communication but also establishes a consistent and reliable cryptographic framework that forms the backbone of secure interactions in IIoT scenarios. 5.