## 5 Performance  
IIoT devices typically face limits in terms of both computing and storage capacity. To determine the efficiency of LI2A, it is necessary to study its storage and computation requirements. This section examines LI2A and four alternative methods [31, 32, 22, 33] with a focus on communication costs, storage requirements, and computing complexity. Table 2 shows the performance evaluation of LI2A, where \"Rand\" represents the invocation of the random number generator and \"hash\" represents the operation of a one-way hash function. Please note that in analyzing the communication cost, we focus on the key agreement stage as it is the primary component of the scheme, in comparison to the online registration phase (which occurs only once and utilizes an engineering computer, as mentioned earlier, mainly involving the \(TA\) adding \(C_{i}\) and \(IIoT_{j}\) to the network) and the FS stage (which requires smaller storage, message size, and computational power). Storage requirement: For each IIoT device, the \(TA\) necessitates storage comprising 256 bits for the shared authentication master key and 32 bits for the corresponding identity \(ID_{j}\). Additionally, \(TA\) also requires equivalent storage to maintain the shared authentication master key and for the user identity \(ID_{i}\). Conversely, the IIoT device or user must store the shared authentication master key \(K_{m}\), session key \(K_{s}\), \(ID_{i}\), and \(ID_{j}\). In total, the IIoT device or user requires a storage capacity totaling \(2*256\) bits + \(2*32\) bits. Communication costs: To achieve the security level corresponding to the 1024-bits RSA algorithm, we represent the lengths of various elements as follows: \(|H|\) for the hash function (256 bits), \(|\oplus K|\) for the message Xored with a key (256 bits), \(|q|\) for a random number (256 bits), \(|ID|\) for an identifier (32 bits), and \(|T|\) for a timestamp (32 bits). In the key agreement phase of the LI2A scheme, \(C_{i}\) initiates the process by sending a message \(m_{3}=(m_{1},m_{2},T_{m_{3}},h(T_{m_{3}}\|K_{s}))\) to \(IIoT_{j}\). \(IIoT_{j}\) responds by sending a message \(m_{4}=(T_{ACK},h(T_{ACK}\|K_{s}))\) to C\({}_{i}\). Consequently, the communication costs are as follows: 800 bits for the user side (\(|\oplus K|+|\oplus K|+|T|+|H|\)), 288 bits for the \(IIoT\) side (\(|T|+|H|\)), and 0 bits for the \(TA\) as it only participates in the online registration. The communication costs are compared to those of other state-of-the-art schemes in Figure 5. It can be seen that LI2A outperforms these schemes in terms of overall communication cost. Computation costs: We assessed the computing costs of the authentication and key agreement phase by calculating the number of significant operations, such as hash functions, scalar multiplications, and fuzzy extractors. To replicate the \(TA\), we utilized the Ali Cloud \begin{table}
\begin{tabular}{l l l} \hline
**Parameter** & **Component** & **Cost** \\ \hline \multirow{3}{*}{Storage} & \(C_{i}\) & 576 bits \\ \cline{2-3}  & \(IIoT_{j}\) & 576 bits \\ \cline{2-3}  & \(TA\) & \(N_{C_{i}}\cdot 288+N_{IIoT_{j}}\cdot 288\) \\ \hline \multirow{3}{*}{Computation} & \(C_{i}\) & 7 hash + 1 Rand / 5 hash + 2 Rand \\ \cline{2-3}  & \(IIoT_{j}\) & 7 hash / 5 hash \\ \cline{2-3}  & \(TA\) & 3 Rand \\ \hline \multirow{3}{*}{Communication} & \(C_{i}\) & 800 bits \\ \cline{2-3}  & \(IIoT_{j}\) & 288 bits \\ \cline{1-1} \cline{2-3}  & \(TA\) & 0 bits \\ \hline \end{tabular}
\end{table} 
Table 2: Performance summary. Figure 5: Communication cost comparison with state-of-the-art schemes [22; 31; 32; 33]. platform, which is equipped with an Intel Xeon CPU E5-2630, 2.30 GHz, 1-GB RAM, and 64-bit Ubuntu 14.04 operating system. A Google Nexus One smartphone with a 2 GHz ARM CPU and Android 4.4.2 was used for simulating user or IoT devices. By utilizing the Miracl library [34], we estimated the duration of execution for operations associated with schemes [22; 31; 32; 33] and LI2A.