## 6 Efficiency Performance
### Cost of Computation  
The cryptographic operations in the schemes of [PERSON] et al. [25] and [PERSON] et al. [26] are built on a bilinear pair, while the schemes of [PERSON] et al. [28], [PERSON] et al. [29] and [PERSON] et al. [31] use ECC. In contrast, our work uses the [PERSON] polynomial-based scheme for resisting side-channel attacks in 5G-enabled vehicular networks. Table 4 presents the comparison of the signing and verifying messages and analyzed schemes comparison in Figure 6.  
Before sending the message in the scheme of [PERSON] et al. [25], the enrolled vehicle requires one Point-to-Map hashing operation to perform the signing message process. Thereby, the entire runtime is \(T_{ptm}\approx 0.937\) ms. Meanwhile, before accepting the message, the enrolled vehicle requires three BPC operations and one Point-to-Map hashing operation to perform the verifying message process. Thereby, the entire runtime is \(3T_{pair-bp}+T_{ptm}\approx 5.548\) ms.  
Before sending the message in the scheme of [PERSON] et al. [26], the enrolled vehicle requires one Point-to-Map hashing operation to perform the signing message process. Thereby, the entire runtime is \(T_{ptm}\approx 0.937\) ms. Meanwhile, before accepting the message, the enrolled vehicle requires three ECC operations to perform the verifying message process. Thereby, the entire runtime is \(3T_{pair-bp}\approx 4.611\) ms.  
Before sending the message in the scheme of [PERSON] et al. [28], the enrolled vehicle requires three ECC operations to perform the signing message process. Thereby, the entire runtime is \(3T_{mul-ccc}\approx 2.145\) ms. Meanwhile, before accepting the message, the enrolled vehicle requires three ECC operations to perform the verifying message process. Thereby, the entire runtime is \(3T_{mul-ccc}\approx 2.145\) ms.  
Before sending the message in the scheme of [PERSON] et al. [29], the enrolled vehicle requires two ECC operations to perform the signing message process. Thereby, the entire runtime is \(2T_{mul-ccc}\approx 1.43\) ms. Meanwhile, before accepting the message, the enrolled vehicle requires four ECC operations to perform the verifying message process. Thereby, the entire runtime is \(4T_{mul-ccc}\approx 2.86\) ms.  
Before sending the message in the scheme of [PERSON] et al. [31], the enrolled vehicle requires three ECC operations to perform the signing message process. Thereby, the entire \begin{table}
\begin{tabular}{c c c} \hline \hline
**Schemes** & **Signing Messages** & **Verifying Messages** \\ \hline [PERSON] et al. [25] & \(T_{ptm}\approx 0.937\) ms & \(3T_{pair-bp}+T_{ptm}\approx 5.548\) ms \\ [PERSON] et al. [26] & \(T_{ptm}\approx 0.937\) ms & \(3T_{pair-bp}+4.611\) ms \\ [PERSON] et al. [28] & \(3T_{mul-ccc}\approx 2.145\) ms & \(3T_{mul-ccc}\approx 2.145\) ms \\ [PERSON] et al. [29] & \(2T_{mul-ccc}\approx 1.43\) ms & \(4T_{mul-ccc}\approx 2.86\) ms \\ [PERSON] et al. [31] & \(3T_{mul-ccc}\approx 2.145\) ms & \(3T_{mul-ccc}\approx 2.145\) ms \\ Our work & \(2T_{chev}\approx 0.682\) ms & \(3T_{chev}\approx 1.023\) ms \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the signing and verifying messages. Figure 6: Analysed schemes comparison. runtime is \(3T_{mul-ccc}\approx 2.145\) ms. Meanwhile, before accepting the message, the enrolled vehicle requires four ECC operations to perform the verifying message process. Thereby, the entire runtime is \(3T_{mul-ccc}\approx 2.145\) ms.  
Before sending the message in our work, the enrolled vehicle requires two Chebyshev polynomial operations to perform the signing message process. Thereby, the entire runtime is \(2T_{chev}\approx 0.682\) ms. Meanwhile, before accepting the message, the enrolled vehicle requires three Chebyshev polynomial operations to perform the verifying message process.