## 5 Numerical Studies
The flow updating model can accommodate such settings. Also, the simulations were conducted on one desktop with one thread, while in real implementation, outputs can be generated parallelly on all participating vehicles, which can reduce the total computational time as well. Given the exponential increase in the MPC scheme computational time with the increase in MPC parties, a potential refinement is to subdivide the collaborative routing problem into smaller portions. This process would enable the MPC schemes to be executed with a reduced number of parties involved in each smaller subproblem. The results obtained from these smaller subproblems can then be aggregated by another MPC scheme. Structuring MPC in this hierarchical way is likely to address the computational issue associated with a large number of MPC parties. Next, the computational feasibility of the route validation scheme is evaluated. The results in Figure 9 seem counter-intuitive at first glance; the time the leader vehicle takes to generate the request, the time that witness vehicles take to reply to the leader, and the total  
Figure 8: MPC output generation times under different \(N\) and \(n_{s}\). Figure 7: Computational times of the collaborative routing scheme. computational time all decrease as the number of witness vehicles increases. Since \(n_{p}\) is fixed, an increase in the number of witness vehicles will result in a decrease in the number of fake signatures that the leader generates; thus, the leader request time is reduced. Also, when witness vehicles generate the replies, the most time-consuming step is constructing the polynomial based on all fake signatures, which takes more time as the number of fake signatures increases. Therefore, the reply times of witness vehicles indicated by the yellow bars in Figure 9 decrease as the number of witness vehicles increases. To evaluate how the value of the privacy protection parameter \(\eta\) influences the efficiency of the route validation scheme, we compare the computational time of simulations with different \(\eta\) when the number of required witness vehicles \(n_{w}=5\). Figure 10 shows that the computational time decreases significantly as \(\eta\) increases, especially the time that witness vehicles take to generate a reply message and the message verification time of \(\mathcal{V}\). It should be noted that the collaborative routing scheme and route validation scheme encompass a variety of critical cryptographic operations, including the EC Elgamal key generation algorithm and signature algorithm. The time complexity of these components is substantially influenced by the parameters of the cryptographic settings, such as the order \(q\) of the addition group \(\mathcal{H}\). This section predominantly focused on analyzing the effects of parameters that hold greater relevance to the transportation context. In practical applications, the selection of these parameters must strike a balance between privacy security and computational efficiency. Securing privacy is certainly a crucial aspect, but the emphasis should equally be on computational feasibility, especially for vehicular applications. Generally, enhancing privacy protection could imply a potential trade-off in computational efficiency. For instance, using a smaller \(\eta\) affords better privacy protection for witness vehicles, as more fake signatures are blended into the position proof. However, this could concurrently increase the computational times, as indicated in Figure 10, thus  
Figure 10: Computational times of the route validation scheme under different values of privacy protection parameter \(\eta\). Figure 9: Computational times of the route validation scheme with different numbers of witness vehicles when \(n_{p}=20\). impacting the route validation scheme's efficiency. Hence, a careful trade-off must be enabled in practice.