## 6 Conclusions  
This work presents the Lightweight Industrial IoT Authentication (LI2A) method, specifically designed to address the security concerns that arise in Industrial Internet of Things (IIoT) contexts. The proposed system model comprises a Trusted Authority (\(TA\)), Users (\(C_{i}\)), and Industrial IoT devices (\(I1 oT_{i}\)), establishing a robust framework for secure communication and authentication. Utilizing tamper-resistant memory for both users and devices ensures the safeguarding of authentication keys and critical data. The proposed LI2A scheme incorporates essential security features, including mutual authentication, authenticated message integrity, key agreement, soundness, forward secrecy, resilience against diverse attacks, and minimal resource overhead. Our comprehensive security analysis demonstrated that LI2A effectively mitigates potential threats, ensuring a secure communication environment for users and IIoT devices. The performance evaluation, which considers storage requirements, communication costs, and computation complexity, has revealed the efficiency of LI2A compared to alternative schemes cited in references [22; 31; 32; 33]. The proposed scheme demonstrated superior communication and computational capabilities, making it particularly well suited for deployment in resource-constrained IIoT devices commonly found in industrial settings. LI2A emerges as a promising and practical solution for securing communications within Industrial IoT environments. Its lightweight design achieves a balance between efficiency and security, offering a robust authentication framework tailored for the dynamic landscape of industrial automation. By addressing the unique challenges posed by IIoT, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **LI2A** & **[22]** & **[31]** & **[32]** & **[33]** \\ \hline \(C_{i}\) & \(7T_{h}+T_{r}/5T_{h}+T_{r}\) & \(6T_{h}+T_{r}\) & \(14T_{h}+T_{f}\) & \(6T_{m}+4T_{a}+5T_{h}\) & \(T_{f}+5T_{m}+T_{a}+16T_{h}\) \\ \hline \(I1 oT_{j}\) & \(7T_{h}/5T_{h}\) & \(6T_{h}/4T_{h}/7T_{h}+T_{r}\) & \(7T_{h}\) & \(T_{e}+4T_{m}+3T_{a}+5T_{h}\) & \(4T_{m}+T_{a}+8T_{h}\) \\ \hline TA & NA & \(7T_{h}/4T_{h}\) & \(9T_{h}\) & \(T_{s}\) & \(2T_{m}+11T_{h}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of cryptographic operations and computational expense. \begin{table}
\begin{tabular}{c c} \hline \hline
**Operation** & **Time (ms)** \\ \hline Hash function (\(T_{h}\)) & 0.03 \\ Random number generator (\(T_{r}\)) & 0.24 \\ Symmetric encryption (\(T_{s}\)) & 0.009 \\ Fuzzy extractor (\(T_{f}\)) & 2.226 \\ Scaler multiplication (\(T_{m}\)) & 2.3 \\ Point addition (\(T_{a}\)) & 0.01 \\ Bilinear pairing (\(T_{e}\)) & 8.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Execution time of various operations. LI2A contributes to the establishment of a secure and reliable foundation for integrating IIoT devices into critical industrial processes. For future work, it is recommended to evaluate LI2A in real-world scenarios by applying it to specific industrial-use cases, such as securing Supervisory Control and Data Acquisition (SCADA) systems or smart manufacturing environments. This testing will provide practical insights into its performance, scalability, and adaptability in addressing sector-specific challenges. Additionally, investigating its integration with advanced IoT ecosystems, including edge computing frameworks and AI-driven predictive analytics, could further enhance its applicability and relevance in modern industrial settings. Conceptualization, A.T.A.G., M.A.M., S.A. and I.T.A. ; methodology, A.T.A.G. and M.A.M. ; validation, S.A. and I.T.A. ; formal analysis, A.T.A.G., investigation, A.T.A.G., M.A.M., S.A. and I.T.A. ; writing--original draft preparation, A.T.A.G., M.A.M., S.A. and I.T.A. ; writing--review and editing, A.T.A.G., M.A.M., S.A. and I.T.A. ; visualization, A.T.A.G., M.A.M., S.A. and I.T.A. ; supervision, A.T.A.G. ; project administration, A.T.A.G. All authors have read and agreed to the published version of the manuscript.