# An ECC-Based Anonymous and Fast Handover Authentication Protocol for Internet of Vehicles
## 6 Performance Analysis
### Computational Cost  
All protocols were achieved on a four-core 3.2 GHz machine with 8 GB memory, and the results were averaged over 300 randomized simulation runs [31]. The execution time of each basic operation is shown in Table 8[29].  
The protocol of [PERSON] et al. [28] requires four one-way hash operations, four bilinear pairing operations, six exponentiation operations, three modular multiplication operations and two modular inverse operations in the initial authentication phase. Therefore, the computational cost of this scheme in the initial authentication phase is about \(4T_{h}+4T_{bp}+6T_{ex}+3T_{mm}+2T_{mi}\approx 292.84\) ms. In the handover authentication phase, it requires two one-way hash operations, two bilinear pairing operations, six exponentiation operations and five modular multiplication operations. Therefore, the computational cost of this scheme in the handover authentication phase is about \(2T_{h}+2T_{bp}+6T_{ex}+5T_{mm}\approx 204.46\) ms.  
The protocol of [PERSON] et al. [29] requires twelve one-way hash operations and six elliptic curve scalar multiplications in the initial authentication phase. Therefore, the computational cost of this scheme in the initial authentication phase is about \(12T_{h}+6T_{m}\approx 106.44\) ms. In the handover authentication phase, it requires twenty-one one-way hash operations and twelve elliptic curve scalar multiplications. Therefore, the computational cost of this scheme in the handover authentication phase is about \(21T_{h}+12T_{m}\approx 211.92\) ms.  
The protocol of [PERSON] et al. [30] requires four one-way hash operations, two elliptic curve scalar multiplications, two bilinear pairing operations and six exponentiation operations in the initial authentication phase. Therefore, the computational cost of this scheme in the initial authentication phase is about \(4T_{h}+2T_{m}+2T_{bp}+6T_{ex}\approx 234.90\) ms. In the handover authentication phase, it requires three one-way hash operations, two elliptic curve scalar multiplications, two bilinear pairing operations and seven exponentiation operations. Therefore, the computational cost of this scheme in the handover authentication phase is about \(3T_{h}+2T_{m}+2T_{bp}+7T_{ex}\approx 253.78\) ms.  
In the protocol proposed in this paper, in the initial authentication stage, thirteen one-way hash operations and sixteen elliptic curve scalar multiplication operations are required, so the computational cost of the scheme in the initial authentication stage is about \(13T_{h}+16T_{m}\approx 277.76\) ms; in the handover authentication stage, only nine one-way hash operations and four elliptic curve scalar multiplication operations are required, so the computational overhead of the scheme proposed in this paper in the handover authentication stage is approximately \(9T_{h}+4T_{m}\approx 71.28\) ms.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Operation** & **Execution Time (ms)** \\ \hline \(T_{h}\) & One-way hash function & 0.32 \\ \(T_{m}\) & Elliptic curve scalar multiplication & 17.10 \\ \(T_{ex}\) & Exponentiation operation & 19.20 \\ \(T_{bp}\) & Bilinear pairing operation & 42.11 \\ \(T_{mm}\) & Modular multiplication operation & 0.88 \\ \(T_{mi}\) & Modular inverse operation & 2.64 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Execution time of basic operations.  
Table 9 and Figure 3 shows that the proposed scheme has extremely superior performance in terms of computational overhead in the handover authentication phase. Compared with the protocols of [PERSON] et al. [28], [PERSON] et al. [29], and [PERSON] et al. [30], the proposed scheme has a lower computational overhead in the handover authentication phase.