## 7 Performance Evaluation  
This section describes performance evaluation of the repaired protocol along with other related protocols [4; 10; 11; 12; 14] in security properties and estimated time. We focus on the security against the anonymity, tracking attack, insider attack, replay attack, impersonation attack, man-in-the-middle attack, mutual authentication and the session key forward secrecy. From Table 1, we see that only the repaired protocol, [PERSON]'s protocol [12] and [PERSON] et al. [14] fulfill all the security properties.  
We analyze the time performance of these protocol by analysis of the core cryptographic operations used in each of them, and then estimate the running time of these protocols by adding the time of executed cryptographic operations. We do not consider the possibility of parallel computation with multi-core technologies since most wearable devices are only single core. Pipelining is also not discussed here since the authentication usually needs to be executed once.  
We consider two possible realizations of an \(\mathcal{SN}\). A sensor device using the MICAz with 4 KB RAM (Crossbow Technology, San Jose, CA, USA) and 7-MHz ATmega128L microcontroller (Microchip Technology Inc, Chandler, AZ, USA) and a smart phone using an iPhone 6s (Apple, Cupertino, CA, USA) with 2 GB RAM ARM (armv8-a) CPU. The data are taken from [13; 24; 25] for the time required on the MICAz while we implement those implementations on a smart phone using the Pairing Based Cryptographic Library [26]. The result is summarized in Table 2.  
Table 3 lists the estimated time of the mentioned protocols, considering the above experimental data. From this table, we may observe that the repaired protocol costs more time than [PERSON]'s protocol [4] as it takes six more hash functions, but costs less time than the other related protocols [10; 11; 12; 14].  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **C1** & **C2** & **C3** & **C4** & **C5** & **C6** & **C7** & **C8** \\ \hline
[10] & Y & Y & N & Y & Y & Y & Y & Y \\
[11] & N & N & Y & Y & Y & Y & Y & Y \\
[12] & Y & Y & Y & Y & Y & Y & Y & Y \\
[4] & N & N & N & Y & N & Y & Y & Y \\
[14] & Y & Y & Y & Y & Y & Y & Y & Y \\ Ours & Y & Y & Y & Y & Y & Y & Y & Y \\ \hline \hline \multicolumn{10}{l}{C1: Provide anonymity;} \\ \multicolumn{10}{l}{C2: Withstand tracking attack;} \\ \multicolumn{10}{l}{C3: Withstand insider attack;} \\ \multicolumn{10}{l}{C4: Withstand repay attack;} \\ \multicolumn{10}{l}{C5: Withstand impersonation attack;} \\ \multicolumn{10}{l}{C6: Withstand man-in-the-middle attack;} \\ \multicolumn{10}{l}{C7: Mutual authentication;} \\ \multicolumn{10}{l}{C8: The session key forward secrecy} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the security properties. Y and N stands for fulfilling and not fulfilling the requirement respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Running Time on a Smartphone** & **Running Time on a MICAz** \\ \hline \(T_{h}\) & Hash function & 0.03 ms & 8 ms [25] \\ \(T_{syn}\) & Symmetric encryption/description operation & 0.12 ms & 3.5 ms [24] \\ \(T_{an}\) & Scalar multiplication over elliptic curves & 20.23 ms & 2450 ms [13] \\ \(T_{bp}\) & Bilinear pairing operation & 25.64 ms & 5320 ms [13] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Computation of the cryptographic operations.