## 7 Performance Analysis
### Computational Costs  
We compared the computational overheads of the proposed scheme with existing works [15; 32; 34; 35; 36; 37; 39]. For this, we referred to an experimental setup using a Raspberry Pi 3 B+ Rev 1.3 equipped with a 64-bit operating system, 1 GB of RAM, and 1.4 GHz quad-core processor, as described in [36]. The experiment measured the average computational time using MIRACL, a widely used open-source cryptographic library written in C/C++, supporting encryption algorithms including ECC. In addition, \(T_{h}\), \(T_{fe}\), \(T_{ccm}\), \(T_{ccm}\), \(T_{sym}\), and \(T_{puf}\) are the execution times for the hash function, fuzzy extractor, ECC point multiplication, ECC point addition, symmetric key encryption, and PUF, respectively. The execution time for the PUF was assumed to be about 0.7 times that of the hash function. The average execution time of each function is indicated in Table 4. We ignore the execution times for the concatenation function and the XOR operation, which are negligible. The computational overhead comparison results are in Table 5 and Figure 8. In our scheme, the user required \(13T_{h}\), the server required \(17T_{h}\), and the sensor required \(7T_{h}+1T_{puf}\). Therefore, the total computational overhead was \(37T_{h}+1T_{puf}\). Although our scheme required marginally more computational resources than those in [15; 39], it achieved significantly enhanced security. [PERSON] et al.'s scheme [39] fails to defend against sensor privileged insider, impersonation, and sensor capture attacks. Similarly, [PERSON] et al.'s scheme [15] is exposed to attacks, including impersonation, stolen mobile device, and insider attacks. Moreover, it does not guarantee user untraceability and mutual authentication. Hence, our scheme is efficient regarding security and the computational cost.  
Figure 7: AVISPA simulation results using OFMC and CL-AtSe.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Scheme** & **User** & **Server** & **Sensor** & **Total Cost (ms)** \\ \hline [PERSON] et al. [32] & \(5T_{h}+2T_{sym}+2T_{ecm}\) & \(5T_{h}+2T_{sym}+2T_{ecm}\) & \(3T_{h}+1T_{ecm}\) & 15.529 \\ [PERSON] et al. [34] & \(9T_{h}+2T_{ecm}+1T_{fe}\) & \(8T_{h}\) & \(4T_{h}+1T_{ecm}\) & 15.641 \\ [PERSON] et al. [35] & \(3T_{h}+2T_{ecm}\) & \(7T_{h}+1T_{ecm}\) & \(4T_{h}+1T_{ecm}\) & 13.478 \\ [PERSON] et al. [36] & \(13T_{h}+4T_{ecm}+1T_{eca}+1T_{fe}\) & \(12T_{h}+6T_{ecm}+2T_{eca}\) & \(9T_{h}+4T_{ecm}+1T_{eca}\) & 44.874 \\ [PERSON] et al. [37] & \(12T_{h}+1T_{fe}\) & \(10T_{h}\) & \(10T_{h}\) & 12.176 \\ [PERSON] et al. [39] & \(10T_{h}\) & \(18T_{h}\) & \(8T_{h}\) & 10.815 \\ [PERSON] et al. [15] & \(4T_{h}\) & \(9T_{h}\) & \(4T_{h}\) & 5.253 \\ Proposed & \(13T_{h}\) & \(17T_{h}\) & \(7T_{h}+1T_{puf}\) & 11.649 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Computational costs.  
Figure 8: Computational costs comparison [15; 32; 34; 35; 36; 37; 39].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\(T_{h}\)** & \(T_{fe}\) & \(T_{ecm}\) & \(T_{eca}\) & \(T_{sym}\) & \(T_{puf}\) \\ \hline
0.309 ms & 2.288 ms & 2.288 ms & 0.016 ms & 0.018 ms & 0.216 ms \\ \hline \hline \end{tabular}
\end{table}
Table 4: Execution time.