## 5 Analyses
### Performance Evaluation
#### 5.3.1 Computation Cost Analysis  
In this section, we give a formal security proof on the proposed certificate-less signature scheme. While using the computation execution times for various dominant time-consuming cryptographic operations summarized in Table 3, we carry out a computation analysis of related CLAS schemes [2; 23; 27; 25; 2] in terms of the three phases of message signing, individual verify and aggregate verify overhead in RSU. The observation is clear that our proposed scheme, ECLAS, has better computation performance to related works from Table 4. In [27], to generate a signature a vehicle carries out three scalar multiplication, \(3T_{c,m}\), over an elliptic curve. This means the computation cost for signing is \(3T_{c,m}\approx 1.326~{}ms\). Whilst for verifying a signature, three bilinear pairings, one scalar multiplication over an elliptic curve and one map-to-point hash function operations, are required. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Operations** & \(T_{bp}\) & \(T_{bp,m}\) & \(T_{bp,sm}\) & \(T_{bp,a}\) & \(T_{H}\) & \(T_{c,m}\) & \(T_{c,sm}\) & \(T_{c,a}\) & \(T_{h}\) \\ \hline
**Times (ms)** & 4.211 & 1.709 & 0.0535 & 0.0071 & 4.406 & 0.4420 & 0.0138 & 0.0018 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Execution Times of Cryptographic Operations. Thus, individual verification needs \(2T_{bp}+T_{e.m}+T_{H}\approx 17.481\ ms\). In aggregate verification phase, three bilinear pairings, \(n\) scalar multiplication over elliptic curve and \(n\) map-to-point hash function operations are required, \(2T_{bp}+nT_{e.m}+nT_{H}\approx 12.633+4.4198n\ ms\). In the proposed ECLAS scheme, for signature generation a vehicle requires two scalar multiplication with respect to elliptic curve and one hash function operation, \(2T_{e.m}+T_{h}\), amounting to the computation load of \(2T_{e.m}+T_{h}\approx 0.8841\ ms\). For individual signature verification, ECLAS, similarly requires two scalar multiplication with respect to elliptic curve and one hash function operation, \(2T_{e.m}+T_{h}\), amounting to the computation load of \(2T_{e.m}+T_{h}\approx 0.8841\ ms\). Whereas for aggregate signature verification, ECLAS requires \(2n\) scalar multiplication with respect to elliptic curve and \(n\) hash function operation, \(2 nT_{e.m}+nT_{h}\), yielding computation cost of \(2 nT_{e.m}+nT_{h}\approx 0.8841n\ ms\). in a similar manner, the computation cost for other relevant comparable schemes [2; 13; 23; 55] can be calculated. Based on the generated summary results of computation cost comparison done in Table 4 and the visual representation done given in Figure 2 we make conclusion on the performance of ECLAS. It is clear that the proposed ECLAS scheme has all over computation efficiency compared to the rest of the scheme except [13], and although it has a slightly lower signing computation overhead it was found to have security flaws in [23], whereas the proposed scheme satisfies the security requirements and withstands KGC escrow property. For simplicity sake, by regarding equal computation capabilities for signing and verifying then we can lump up the computation load that is incurred in message signing and individual verifying for a single signature. As such, the overall load for [PERSON] et al. [27] comes up to \((1.326+17.481)\ ms=18.807\ ms\) and for [PERSON] et al. [13] the overall load is \((0.4439+1.3298)\ ms=1.7737\ ms\). Proceeding in this manner for the rest of the schemes, in [PERSON] et al. [55], [PERSON] et al. [2], [PERSON] et al. [23] the overall computation loads are; 24.3675 \(ms\), 19.664 \(ms\), 2.1887 \(ms\) respectively.