# 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
 \tag{15}\]  
where \(d_{\text{AS},1}\) and \(d_{\text{SB},1}\) represent the paths from satellite S to ground terminal A and from satellite S to spoofing terminal B in the first TWSTT, respectively. Similarly, \(d_{\text{AS},2}\) and \(d_{\text{SB},2}\) represent the paths from satellite S to ground terminal A and from satellite S to spoofing terminal B in the second TWSTT, respectively. If the spoofing terminal is directly positioned above the ground terminal, the spatial propagation loss of the spoofing signal would be minimized, making it more effective in suppressing the genuine signal. Consequently, this method is considered the most cost-effective way for the spoofing terminal to carry out its deception. To facilitate a more accurate modeling of this scenario, we can establish a spatial geometry model as depicted in Figure 6.  
In the geometric relationship shown in Figure 6, we denote \(H\) as the vertical distance between the LEO satellite and the user terminal, and \(h\) as the vertical distance between the spoofing terminal and the user terminal. Based on this geometry, we can derive the following relationship, where c represents the speed of light. \[\left\{ \begin{array}{ll}cd_{\text{AS},1}=H/\sin\theta_{1}\\ c^{2}d_{\text{AS},1}^{2}+h^{2}-2 hcd_{\text{AS},1}\cos\beta_{1}=c^{2}d_{\text{SB},1}^{2}\\ \theta_{1}+\beta_{1}=90^{\circ}\\ cd_{\text{AS},2}=H/\sin\theta_{2}\\ c^{2}d_{\text{AS},2}^{2}+h^{2}-2 hcd_{\text{AS},2}\cos\beta_{2}=c^{2}d_{\text{ SB},2}^{2}\\ \theta_{2}+\beta_{2}=90^{\circ}\\ \end{array} \right. \tag{16}\]  
Substituting Equation (16) into Equation (15) transforms the expression as follows:  
\[\Delta T_{\text{delay}}=\frac{\left(\frac{H}{\sin\theta_{2}}-\frac{H}{\sin \theta_{1}}\right)}{2c}-\frac{\left(\sqrt{\left(\frac{H}{\sin\theta_{2}} \right)^{2}+h^{2}-2 Hh}-\sqrt{\left(\frac{H}{\sin\theta_{1}}\right)^{2}+h^{2}- 2 Hh}\right)}{2c} \tag{17}\]  
Figure 6: Spatial geometry model in the presence of spoofing signals. It can be seen from Equation (17) that \(\Delta T_{\text{delay}}\) is a function using \(H\), \(h\), \(\theta_{1}\), and \(\theta_{2}\). Where \(h\) is determined by the spoofing terminal, for example, the UAV flight altitude is usually 1-15 km, and \(H\), \(\theta_{1}\), and \(\theta_{2}\) are determined by the satellite position and the user terminal position; when the user terminal is fixed, the values of \(\theta_{1}\) and \(\theta_{2}\) can be changed by changing the moment of initiating the authentication. Therefore, it is necessary to analyze the optimal elevation angle and initiation timing for authentication at the user terminal. By performing calculations, Figure 7 shows the relationship between \(\Delta T_{\text{delay}}\) and \(\theta_{1}\) with \(\theta_{2}\) when different combinations of \(H\) and \(h\) are used. The visible elevation angle of the satellite is generally required to be larger than \(15^{\circ}\). To ensure adequate visibility between the satellite and the user terminal, the range of \(\theta_{1}\) and \(\theta_{2}\) in the figure is set from \(30^{\circ}\) to \(90^{\circ}\). From Figure 7, we can draw the following two conclusions:  
1. When the vertical distance between the LEO satellite and the user terminal \(H\) and the vertical distance between the spoofing terminal and the user terminal \(h\) are fixed, and the elevation angles (\(\theta_{1}\) and \(\theta_{2}\)) of the user terminal to the satellite are equal, it results in \(\Delta T_{\text{delay}}\) being 0. This indicates that the user terminal can successfully be spoofed by a spoofing terminal. 2. The larger the difference between \(\theta_{1}\) and \(\theta_{2}\), the greater the value of \(\Delta T_{\text{delay}}\). In this context, a higher value of \(\Delta T_{\text{delay}}\) indicates a lower likelihood of being spoofed by the spoofing terminal.