## 6 Performance Evaluation
### Computation Cost Analysis and Comparison
Consequently, the total run time is \(3T_{bp}+(2n)T_{bp-pm}+(4n-3)T_{bp-pa}+(n+1)T_{M.T.P}+(2n)T_{h}\)\(\approx\) ms.  
As for the ECC adopted in the proposed scheme, the single message signing process requires registered participating vehicle to run one operation of point addition \(1T_{cc-pa}\), and two operations of hash function \(2T_{h}\). Consequently, the total run time is \(1T_{ccc-pm}+2T_{h}\)\(\approx\) ms. For a single message verification, the verifying recipient must perform two operations of scalar multiplication \(2T_{ccc-pm}\), one operation of point addition \(1T_{ccc-pa}\), and two operations of hash function \(2T_{h}\). Consequently, the total run time is \(2T_{ccc-pm}+1T_{ccc-pa}+2T_{h}\)\(\approx\) ms. To verify multiple messages in the proposed scheme, the verifying recipient needs to carry out two scalar multiplication operations \(2T_{ccc-pm}\), \((n+1)\) point addition operations \((n+1)T_{ccc-pa}\), and \((2n)\) hash function operations \((2n)T_{h}\). Consequently, the total run time is \(2T_{ccc-pm}+(n+1)T_{ccc-pa}+(2n)T_{h}\)\(\approx\) ms.  
As presented in Figure 4, the proposed scheme achieved a much lower computation cost for signing and verifying a single message than the existing schemes. Furthermore, Figure 5 shows that the proposed scheme has a significant advantage in batch verifi  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & **Single Message Singing** & **Single Message Verification** & **Batch Messages Verification** \\ \hline [PERSON] et al. [25] & \( \begin{matrix}3T_{bp-pm}+2T_{bp-pa}+\\ 1T_{M.T.P}+1T_{h}\approx 23.192\text{ ms}\end{matrix} \) & \( \begin{matrix}3T_{bp}+2T_{bp-pm}+1T_{bp-pa}+\\ 2T_{M.T.P}+1T_{h}\geq 23.192\text{ ms}\end{matrix} \) & \( \begin{matrix}3T_{bp}+(2n)T_{bp-pm}+(4n-3)T_{bp-pa}+(n+1)T_{M.T.P}+\\ 27.3644\text{ ms}\end{matrix} \) & \( \begin{matrix}3T_{bp}+(2n)T_{bp-pm}+(4n-3)T_{bp-pa}+(n+1)T_{M.T.P}+\\ (2n)T_{h}\approx 21.5736+7.3476n\end{matrix} \) \\ [PERSON] et al. [26] & \( \begin{matrix}2T_{bp}+5T_{bp-pm}+2T_{hp-pa}\approx\\ 4T_{bp-pm}+2T_{h}\approx 6.2636\text{ ms}\end{matrix} \) & \( \begin{matrix}2T_{bp}+5T_{bp-pm}+(n+1)T_{bp}+(5n)T_{bp-pm}+\\ 19.4702\text{ ms}\end{matrix} \) & \( \begin{matrix}(n+1)T_{bp}+(5n)T_{bp-pm}+\\ (2n)T_{bp-pa}\approx\\ 5.811+13.6592n\end{matrix} \) \\ [PERSON] et al. [29] & \( \begin{matrix}6T_{bp-pm}+1T_{bp-pa}+1T_{h}\approx\\ 9.404\end{matrix} \) & \( \begin{matrix}3T_{bp}+2T_{bp-pm}\approx 20.5638\end{matrix} \) & - \\ [PERSON] et al. [31] & \( \begin{matrix}7T_{ccc-pm}+6T_{h}\approx 4.7086\text{ ms}\end{matrix} \) & \( \begin{matrix}12T_{ccc-pm}+8T_{ccc-pa}+8T_{h}\approx\\ 8.0884\text{ ms}\end{matrix} \) & \( \begin{matrix}(4n+10)T_{ccc-pm}+(6n+2)T_{h}\approx\\ 2T_{ccc-pa}+(6n+2)T_{h}\approx\\ 6.7262+2.718\text{ ms}\end{matrix} \) \\ [PERSON] et al. [32] & \( \begin{matrix}1T_{ccc-pm}+2T_{h}\approx 0.6729\text{ ms}\end{matrix} \) & \( \begin{matrix}4T_{ccc-pm}+1T_{ccc-pa}+2T_{h}\approx\\ 2.6923\text{ ms}\end{matrix} \) & \( \begin{matrix}(2n+2)T_{ccc-pm}+(n)T_{ccc-pa}+(n)T_{h}\approx\\ 1.3436+1.3487n\text{ ms}\end{matrix} \) \\ [PERSON] et al. [36] & \( \begin{matrix}2T_{ccc-pm}+2T_{h}\approx 1.3456\text{ ms}\end{matrix} \) & \( \begin{matrix}3T_{ccc-pm}+2T_{ccc-pa}+2T_{h}\approx\\ 2.0236\text{ ms}\end{matrix} \) & \( \begin{matrix}(n+2)T_{ccc-pm}+(n)T_{ccc-pa}+(2n)T_{h}\approx\\ 2.0236\text{ ms}\end{matrix} \) \\  
cation of multiple messages compared with the eight other authentication schemes of [PERSON] et al. (2015), [PERSON] et al. (2016), [PERSON] et al. (2019), [PERSON] et al. (2019), [PERSON] et al. (2019), [PERSON] et al. (2019), [PERSON] et al. (2019), and [PERSON] et al. (2019). The proposed scheme achieves the best performance among the schemes compared.