# Secure and Efficient High Throughput Medium Access Control for Vehicular Ad-Hoc Network
## 3 Secure VANET Communication (SVC) Using Commutative RSA Technique
If there is no loss, consider the arbitrary back-off time of \(r\) vehicles (arbitrary back-off time of \(r\) vehicles is the random time selected by a set of \(r\) vehicles for contention (that is, \(r\) number of vehicles waiting for a while for channel access after detection of the first collision)) are arranged as \(g_{1}\leq g_{2}\leq\cdots\leq g_{r}\). (\(g_{r}\) is the random backoff time of each participating user in channel contention; \(r\) is the participation number of vehicles in the network. The first vehicle has the least waiting time, and the last vehicle has the maximum waiting time). Suppose \(r\) vehicles/devices in the contention stage; the probability of the 1 st collision is shown as follows:  
\[\sum_{x}^{R}e_{xy}=1\ \forall y \tag{4}\]2  
Each component in the double heaps shows the probability of a collision if \(y\) vehicles choose the same correction value for \(x\). However, the probability of the first collision is computed as follows:  
\[\sum_{x}^{R}e_{xy}=1\ \forall y \tag{4}\]3  
where \(\mathbb{L}\{r\;vehicle/device\;contend\}\) are the possibility of participating in the vehicles \(r\) in the contention stage and the Equation (19) is used to calculate \(\mathcal{L}_{u}^{(r)}\). To rate \(\mathcal{L}_{u}\), we derive \(\mathbb{L}\{r\;vehicle/device\;contend\}\). If we have access to one channel \(T_{x}\) and all channels \(T_{x}^{S}\)occupied, we can prove that \(x\) vehicle will participate in the contention. The probabilities of this scenario are expressed as:  
\[\mathcal{L}_{S}^{(x)}=\mathbb{L}\bigg{\{} \begin{array}{l}1\;channel\;at\;T _{x}^{s}\;is\;accessible\;and\\ \;all\;channels\;at\;T_{x}\;occupied\end{array} \bigg{\}}=\left(\prod_{y\in T_{x}} \bar{l}_{xy}\right)\bigg{(}1-\prod_{y\in T_{x}^{s}}\bar{l}_{xy}\bigg{)} \tag{21}\]  
The probability of the scenario in which the \(r\) vehicles users in the contention phase (contention phase is the period of the request of a channel for data transmission) is  
\[\mathbb{L}\{\;vehicle/device\;r\;contend\}=\sum_{k=1}^{U_{R}^{r}}\prod_{x \in\wedge_{k}}\mathcal{L}_{S}^{(x)}\prod_{x\in\wedge_{k}\wedge_{l}}\mathcal{L }_{S}^{(x)} \tag{22}\]  
\(\wedge_{R}\) is the group of all vehicles \(R\) (\(\{1,2,\ldots;R\}\)), \(\wedge_{k}\) is a particular group of \(r\) users. Output substitution of Equation (22) into Equation (20); \(\mathcal{L}_{u}\) can be calculated. Nevertheless, it becomes possible to define \(\mathcal{A}\) as  
\[\mathcal{A}=\min\{\mathcal{A}|\mathcal{L}_{u}(\mathcal{A})\leq eL\} \tag{23}\]  
where, \(\mathcal{L}_{u}\) in Equation (20) denotes a function of \(\mathcal{A}\). (f) Computation of Mac overhead:  
Equation (23) can be used to model the overhead mean of MAC protocol. Let us consider \(h\) as the average value of the back-off parameters considering the security/safety selected by vehicles. Thus, \(h=\frac{(\mathcal{A}-1)}{2}\), where the back-off value is determined uniformly between \(\mathcal{A}-1\) and \(0\) periods. Average overhead is calculated as follows:  
\[\mathcal{D}(\mathcal{A})=\frac{[\mathcal{A}-1]\varphi/2+s_{CTS}+s_{RTS}+3s_{ SIFS}+s_{SYNC}+s_{SEN}}{S_{\mathcal{A}}} \tag{24}\]where \(s_{CTS}\), \(s_{RTS}\) and \(s_{SIFS}\) are the time corresponding of Request to Send \((RTS)\), Clear to Send \((CTS)\), and Short inter-frame space \((SIFS)\) packets; \(s_{S\text{VNC}}\) is the synchronization of size packets; \(s_{SEN}\) is the sensing time; \(S_{\mathcal{I}}\) is the time cycle, and \(\varphi\) is the one back-off param of corresponding time. The \(\mathcal{D}\). overhead depends on the results of channel allocation. Thus, \(\mathcal{D}\) is updated in flow diagram 2 based on the current channel assigned. Our PECA minimizes collision and maximizes system throughput, and provisioning security to S-PECA does not incur much overhead as proved in the next section experimentally.