## 6 Performance Evaluation
### Computation Cost Analysis and Comparison  
This section discusses [PERSON] et al. 's scheme [25] and the proposed scheme, whereas the schemes of [PERSON] et al. [26], [PERSON] et al. [29], [PERSON] et al. [31], [PERSON] et al. [32], [PERSON] et al. [36], [PERSON] et al. [39], and [PERSON] et al. [38] are analyzed using the same method. Table 4 presents the computation cost of each process. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Scheme** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **Proposed** \\  & [25] & [26] & [29] & [31] & [32] & [36] & [39] & [38] & Scheme \\ \hline Traceability and & & & & & & & & & \\ Revocability & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ Identity & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ privacy-preserving & & & & & & & & & \\ Message integrity & & & & & & & & & \\ and authentication & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ Unlinkability & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ No RSU aided & ✗ & ✗ & ✗ & ✗ & ✗ & ✗ & ✓ & ✓ & ✓ \\ No TPD aided & ✗ & ✗ & ✗ & ✗ & ✗ & ✗ & ✗ & ✗ & ✓ \\ Resistance to & & & & & & & & & \\ impersonation & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ attack & & & & & & & & & \\ Resistance to & & & & & & & & & \\ modify attack & & & & & & & & & \\ Resistance to & & & & & & & & & \\ password-guessing & ✗ & ✗ & ✗ & ✗ & ✗ & ✗ & ✓ & ✓ & ✓ \\ attack & & & & & & & & & \\ Resistance to & & & & & & & & & \\ replay attack & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Security comparison. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Cryptographic Operation** & **Notation** & **Run Time (Milliseconds)** \\ \hline The bilinear pairing operation & \(T_{bp}\) & 5.811 \\ The scalar multiplication operation of the bilinear pairing & \(T_{bp-pm}\) & 1.5654 \\ The point addition operation of the bilinear pairing & \(T_{bp-pa}\) & 0.0106 \\ The MapToPoint hash function operation & \(T_{M.T.P}\) & 4.1724 \\ The scalar multiplication operation operation of ECC & \(T_{ecc-pm}\) & 0.6718 \\ The point addition operation & \(T_{ecc-pa}\) & 0.0031 \\ The secure cryptographic hash function operation & \(T_{h}\) & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The run times of cryptographic operations. The scheme of [PERSON] et al. [25] relies on bilinear pairing operations. A single message signing process in [PERSON] et al. 's scheme [25] requires a registered participating vehicle to run three operations of scalar multiplication \(3T_{bp-pm}\), two operations of point addition \(2T_{bp-pa}\), one MapReducePoint hash function operation \(1T_{M.T.P}\), and one operation of hash function \(1T_{h}\). Consequently, the whole run time is \(3T_{bp-pm}+2T_{bp-pa}+1T_{M.T.P}\)\(+\)\(1T_{h}\)\(\approx\) ms. A single message-verification process in [PERSON] et al. 's scheme [25] requires the verifying recipient to perform three operations of bilinear pairing \(3T_{bp}\), two operations of scalar multiplication \(2T_{bp-pm}\), one operation of point addition \(1T_{bp-pa}\), two operations of MapToPoint hash function \(2T_{M.T.P}\), and one operation of hash function \(1T_{h}\). Consequently, the total run time is \(3T_{bp}+2T_{bp-pm}+1T_{bp-pa}+2T_{M.T.P}+1T_{h}\)\(\approx\) ms. The process of verifying multiple messages in [PERSON] et al. 's scheme [25] requires the verifying recipient to run three bilinear pairing operations \(3T_{bp}\), \((2n)\) scalar multiplication operations \((2n)T_{bp-pm}\), \((4n-3)\) point addition operations \((4n-3)T_{bp-pa}\), \((n+1)\) MapToPoint hash function operations \((n+1)T_{M.T.P}\), and \((2n)\) hash functions operations \((2n)T_{h}\).