# CAVVPM: Challenge-Based Authentication and Verification of Vehicle Platooning at Motorway
## 4 Security Analysis
### Remote Adversary
#### 4.2.2 A Vehicle Follow Verifier  
We take into account the scenario in which a vehicle \(R\) other than the adversary follows the verifier. A safe distance \(d_{\mathit{ref}}\) between the adversary's opponent and the verifier's vehicle \(R\) is maintained, but it is not under the control of the adversary. A requesting message \(\mathrm{m_{M}}\) (1) is sent to the validator by the malicious user asking to join the platoon. The enemy will be able to pass the electronic identity verification, as previously stated. With help from a series of demands \(\Gamma\), the verifier will confront the opponent. Instead of the distant opponent, the verifier will estimate the distance to the following vehicle \(R\) at the deadlines stated in \(\Gamma\). In order for the PoF to be passed, the opponent must be at the milestones by the appropriate dates set by \(R\). **Using a random walk to model [PERSON]'s path**: [PERSON]'s random walk around \(d_{\mathit{ref}}\) serves as the basis for calculating [PERSON]'s chance of passing the PoF test. While still following, [PERSON] may move independently of the platoon and change its distance from it within a certain range. Under a certain range \([d_{\mathit{min}},\,d_{\mathit{max}}]\), the vehicle R varies its range from the verifier. N-state Markov chains are used to simulate [PERSON]'s random walk where states reflect the possible locations of [PERSON] and transition probabilities indicate the likelihood of moving to a new position within the range after a time step of n. As a result, we limit the range \([d_{\mathit{min}},\,d_{\mathit{max}}]\), In this case, we assume that [PERSON] can traverse a specified distance \(d_{\mathit{step}}\) during a defined time step, and divide the range by \(d_{\mathit{step}}\) to obtain a total of N places (states). The initialization allocation \(\mathrm{P^{(0)}}\) at time 0 is considered to be regular without losing clarity. The odds of changing states are also provided by a \(N\times N\) matrix \(P=\left(P_{i_{j}}\right)\) with:  
\[P_{1,1}=P_{1,2}=P_{N,N}=P_{N,N-1}=1/2 \tag{6}\]  
\[P_{i,i+1}=P_{i,i-1}=P_{i,i}=1/3,i=2\ldots N-1 \tag{7}\]  
\[P_{i,j}=0,\,\,\text{all other}\,\,i,j\text{:} \tag{8}\]  
Figure 6 depicts the randomized walk's potential range (a). There will always be a transit to a new state in a random stroll, and we have decided to allow the vehicle to remain in the same condition within such a time interval. Furthermore, given a state, the probability of moving ahead, backward, or staying in the same shape is equal, regardless of the value of matrix P. The M potential checkpoints specified by the verifier might not even correspond with the N candidate states of R. However, we may suppose that R's state  
space includes M checkpoints. To use the random walk model, we can now calculate the likelihood of clearing the PoF validation, disregarding the beginning and end positions of \(d_{\text{ref}}\). **Proposition 1**. : _Let the verifier put \(M\) through a series of K tasks \(\Gamma=\{(d_{1},t_{1}),(d_{2},t_{2}),\ldots(d_{K},t_{K})\}\). In a state space \(S\) of size \(M\), each milestone is randomly chosen. A nonlinear model using \(S\subseteq S^{\prime}\) may be used to find a state space \(S^{\prime}\) of size \(N\) and a vehicle \(R\) that follows the verifier. Due to K's movement, the chance that \(M\) passes the PoF verification is provided by:_  
\[\mathbf{P}_{M}=\left(\frac{1}{NM}\right)^{K}\prod_{k=1}^{K}\sum_{i=1}^{M}\sum _{j=1}^{N}P_{j,i}^{\sum_{k=1}^{k}n_{k}}. \tag{9}\]  
Following \(n\) rounds, the transition probability matrix \(P^{n}\) is shown in the following figure. **Evidence: Passing a single challenge:** Checkpoints and deadlines may be combined into one challenge (d, t). If R is at a distance d at time t, the opponent has won the challenge. Probabilities (X1, X2, etc.) are used to represent R's non-linear model and the nth simulation time of the discrete stochastic process (DSP).