# A NavCom Signal Authentication Scheme Based on Twice Two-Way Satellite Time Transfer
## 2 A Twice TWSTT NavCom Signal Authentication Scheme
### Signal and Detection Model
Therefore, in order to ensure that this authentication scheme can perform optimally, the moment when the user terminal initiates the authentication request needs to meet the requirement of maximizing the difference between the values of \(\theta_{1}\) and \(\theta_{2}\). Without loss of generality, we set \(\theta_{1}=30^{\circ}\), \(\theta_{2}=90^{\circ}\) or \(\theta_{1}=90^{\circ}\), \(\theta_{2}=30^{\circ}\). As users can obtain satellite ephemeris through communication signals in advance, they can predict the launch angle and launch time in advance. Figure 8 illustrates the variation trend of \(\Delta T_{\text{delay}}\) versus \(h\) and \(H\) when \(\theta_{1}=30^{\circ}\) and \(\theta_{2}=90^{\circ}\). From Figure 8, we can draw the following conclusions:  
1. When \(\theta_{1}\), \(\theta_{2}\), and \(H\) are fixed during user terminal authentication, it can be observed that the smaller the \(h\), the smaller \(\Delta T_{\text{delay}}\) becomes, resulting in a higher likelihood of successful spoofing. 2. When \(\theta_{1}\), \(\theta_{2}\), and \(h\) are fixed, the larger \(H\) and the smaller \(\Delta T_{\text{delay}}\) become, the easier it is to spoof successfully. \(H\) has a relatively small effect on the \(\Delta T_{\text{delay}}\). In contrast, \(h\) has a larger effect on the \(\Delta T_{\text{delay}}\). To analyze the impact of \(h\) and \(H\) on \(\Delta T_{\text{delay}}\) more effectively, we substitute the optimal elevation angles into Equation (17); \(\Delta T_{\text{delay}}\) can be rewritten as follows:  
\[\Delta T_{\text{delay}}\ =\frac{2H-\sqrt{4H^{2}+h^{2}-2 Hh}-h}{2c} \tag{18}\]  
Owing to the user terminal already possessing the satellite ephemeris in advance, the value of \(H\) is known to the user terminal. Therefore, we can consider \(\Delta T_{\text{delay}}\) as a function solely dependent on \(h\). Through numerical calculations and fitting analysis, we establish the following relationship:  
\[ \begin{array}{c}\Delta T_{\text{delay}}\ (h)\ \sim U\big{\{}4.16\times 10^{-22}h^{2}-1.22\times 10^{-15 }h+8.34\times 10^{-7},\\ 9.48\times 10^{-20}h^{2}-2.77\times 10^{-13}h+1.28\times 10^{-5}\big{\}}\end{array}  \tag{19}\]  
The fitted correlation coefficient of Equation (19) is 0.9923, indicating a strong correlation between the variables \(h\) and \(\Delta T_{\text{delay}}\). Additionally, the mean squared error of the fitting is 1.294 ps, suggesting that the Equation (19) provides an accurate characterization of the original function. To investigate the performance limits of the proposed NavCom signal authentication scheme, the worst-case spoofing scenario is considered. In this scenario, we assume that \(\Delta T_{\text{delay}}\) takes its minimum value, and then the Equation (14) can be transformed into the following:  
\[ \begin{array}{c}\text{H}_{0}:R_{\text{outh}}\ \sim\text{N}\Big{(}\min \Big{(}\Delta T_{\text{delay}}\ \Big{)},2T_{C,\text{A}}^{2}\sigma_{IDLL,\text{A}}^{2}+2T_{C,S}^{2} \sigma_{IDLLS}^{2}\Big{)}\end{array}  \tag{20}\]  
where \(\min\Big{(}\Delta T_{\text{delay}}\Big{)}=4.16\times 10^{-22}h^{2}-1.22\times 10^{- 15}h+8.34\times 10^{-7}\). As a result, the hypothesis testing for the certified detection volume of this method is shown below:  
\[\begin{array}{c}\text{H}_{0}:R_{\text{outh}}\ \sim\text  
When the false alarm probability of the constant false alarm detection is denoted as \(P_{FA}\), the detection threshold \(R_{\text{th}}\) can be determined by Equation (22). \[ \begin{array}{ll}R_{\text{th}}(P_{FA})&=\text{min}\Big{(}\Delta T_{\text{delay }}\Big{)}\\ &-\sqrt{2T_{\mathcal{C},\mathcal{A}}^{2}\sigma_{IDLL,\mathcal{A}}^{2}+2T_{ \mathcal{C},\mathcal{S}}^{2}\sigma_{IDLL,\mathcal{S}}^{2}}Q^{-1}(P_{FA})\end{array}  \tag{22}\]  
where \(Q(x)=\int_{x}^{+\infty}\frac{1}{\sqrt{2\pi}}\exp\Bigl{(}-\frac{1}{2}t^{2} \Bigr{)}dt\) is the right-tailed function of the standard normal distribution. At this point, the detector for signal authentication can be defined as Equation (23). \[