# 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, when there is no repeater spoofing interference signal, it can be obtained that the measurement errors of \(\mathrm{TIC(A)}\) and \(\mathrm{TIC(S)}\) obey the probability density distributions shown in Equation (8). \[\mathrm{TIC}(A) \sim\mathrm{N}\Big{(}0,T_{C,A}^{2}\sigma_{IDLLA}^{2}\Big{)} \tag{8}\] \[\mathrm{TIC}(S) \sim\mathrm{N}\Big{(}0,T_{C,S}^{2}\sigma_{IDLLS}^{2}\Big{)}\]  
where \(T_{C,A}\) and \(\sigma_{IDLLA}\) are the pseudo-code width and mean square deviation of the downlink signal's code phase measurement error, and \(T_{C,S}\) and \(\sigma_{IDLLS}\) are the pseudo-code width and mean square deviation of the uplink signal's code phase measurement error, respectively. Then, the clock difference measurement \(\Delta T\) will obey the probability density distribution shown in Equation (9), where \(T\) is the true value of the clock difference. \[\Delta T\sim\mathrm{N}\Big{(}T,T_{C,A}^{2}\sigma_{IDLLA}^{2}+T_{C,S}^{2}\sigma _{IDLLS}^{2}\Big{)} \tag{9}\]  
When there is no repeater spoofing interference signal, the two clock difference measurements \(\Delta T_{1}\) and \(\Delta T_{2}\) obey the probability density distribution shown in Equation (10). \[\Delta T_{1} \sim\mathrm{N}\Big{(}T,T_{C,A}^{2}\sigma_{IDLLA}^{2}+T_{C,S}^{2} \sigma_{IDLLS}^{2}\Big{)} \tag{10}\] \[\Delta T_{2} \sim\mathrm{N}\Big{(}T,T_{C,A}^{2}\sigma_{IDLLA}^{2}+T_{C,S}^{2} \sigma_{IDLLS}^{2}\Big{)}\]  
For further analysis, we define the certified detection volume of this method as the difference between two clock difference measurements, as shown in Equation (11). \[\sigma_{IDLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}D\bigg{(}1+\frac{2}{(2-D)T_{ \mathrm{coh}}\ \cdot C/N_{0}}\bigg{)}} \tag{6}\]0  
Therefore, when there is no repeater spoofing interference signal, the authentication detection volume will obey the probability density distribution shown in Equation (12). \[\sigma_{IDLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}D\bigg{(}1+\frac{2}{(2-D)T_{ \mathrm{coh}}\ \cdot C/N_{0}}\bigg{)}} \tag{6}\]1  
When the repeater spoofing interference signal is present, there will be an offset in the clock difference due to the spoofing signal, which can be denoted as \(T_{\mathrm{delay}}\), and \(T_{\mathrm{delay}}=\frac{d_{AS}-(d_{BS}+d_{B}+d_{B})}{2}\). At this point, the two clock difference measurements \(\Delta T_{1}\) and \(\Delta T_{2}\) will obey the probability density distribution shown in Equation (13). \[\sigma_{IDLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}D\bigg{(}1+\frac{2}{(2-D)T_{ \mathrm{coh}}\ \cdot C/N_{0}}\bigg{)}} \tag{6}\]2 \[\sigma_{IDLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}D\bigg{(}1+\frac{2}{(2-D)T_{ \mathrm{coh}}\ \cdot C/N_{0}}\bigg{)}} \tag{6}\]3  
where \(T_{\mathrm{delay},1}\) and \(T_{\mathrm{delay},2}\) represent the clock difference offsets of the twice TWSTT, respectively. Therefore, when the repeater spoofing interference signal is present, the authentication detection volume will obey the probability density distribution shown in Equation (14). \[\sigma_{IDLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}D\bigg{(}1+\frac{2}{(2-D)T_{ \mathrm{coh}}\ \cdot C/N_{0}}\bigg{)}} \tag{6}\]4  
where \(\Delta T_{\mathrm{delay}}=T_{\mathrm{delay},1}-T_{\mathrm{delay},2}\). Owing to \(\Delta T_{\mathrm{delay}}\) not being a fixed value, its probability distribution must be analyzed. Due to once TWSTT time being relatively short and the ground terminal being fixed in most scenarios, it is assumed that the positions of the spoofing terminal and the ground terminal remain unchanged during the twice TWSTT processes, then \(\Delta T_{\text{delay}}\) can be rewritten as Equation (15). \[ \begin{array}{ll}\Delta T_{\text{delay}}&=\frac{d_{\text{AS}2}-(d_{\text{SB}2} +d_{\text{B}}+d_{\text{BA}})}{2}-\frac{d_{\text{AS}1}-(d_{\text{SB}1}+d_{\text {B}}+d_{\text{BA}})}{2}\\ &=\frac{(d_{\text{AS}2}-d_{\text{AS}1})-(d_{\text{SB}2}-d_{\text{SB}1})}{2} \\ \end{array}