## 3 CV QKD over Satellite Channels
### Satellite-to-Ground Channel Model  
In our study we consider that downlink satellite channels, as the turbulence effects, being destructive for CV QKD [48], are less pronounced in this regime compared to the uplink scenario, where the signal is strongly affected by the atmosphere already in the beginning of the channel [54]. Optical Gaussian beam in the satellite-to-ground link is influenced by analytically predictable systematic and statistical effects occurring during the communication window. Aside from losses within the receiver optical system \(\eta_{det}\), due to coupling and detection inefficiencies, systematic effects also include diffraction and refraction induced losses. The main source of loss is beam-spot broadening caused by diffraction. The broadening limits the maximal transmission efficiency \(\eta_{0}\) as finite receiving aperture of size \(a\) truncates and measures only a part of the incoming collimated beam with \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Coherent States**} & \multicolumn{3}{c}{**Squeezed States**} \\  & **100 MHz** & **GHz** & **10 GHz** & **100 MHz** & **GHz** & **10 GHz** \\ \hline Active collective attack & 23 dB & 24 dB & 25 dB & 29 dB & 30 dB & 31 dB \\ Passive collective attack & 27 dB & 32 dB & 37 dB & 33 dB & 37 dB & 42 dB \\ Active individual attack & 29 dB & 34 dB & 39 dB & 33 dB & 37 dB & 43 dB \\ Passive individual attack & 29 dB & 34 dB & 39 dB & 33 dB & 38 dB & 43 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tolerable levels of channel attenuation (rounded) for various security assumptions and optimized CV QKD protocol settings. Excess noise on the channel output is fixed to \(\epsilon=10^{-4}\) SNU and optimal squeezing is \(V_{S}\geq 0.1\) SNU. During a passive attack the excess noise is presumed to be trusted. spot-size \(W\). The latter is largely determined as \(W=\theta_{d}L(\zeta)\) by divergence of the beam \(\theta_{d}\) and the channel length \(L(\zeta)\). Line-of-sight distance, also referred to as slant range, between the observer and the satellite depends on the exact position of the latter. In following we assume a perfectly circular Low-Earth orbit within observers meridian plane, characterized by the altitude above the ground \(H\), ranging from 200 km to 2000 km. The slant range is obtained as  
\[L(\zeta)=\sqrt{H^{2}+2 HR_{\oplus}+R_{\oplus}^{2}\cos^{2}\zeta}-R_{\oplus}\cos\zeta, \tag{5}\]  
where \(R_{\oplus}\) is Earth radius and \(\zeta\) is zenith angle i.e., between the line pointing in opposite direction to the gravity from the observer and the slant range, see also Figure 2. Additionally, air density and optical refractive index change with the altitude causing the bending of the beam and making the overall traveling distance longer than the straight geometrical slant range. The refraction induced elongation also depends on the signal wavelength, geographical position and altitude of the receiver as well as atmospheric conditions (relative humidity, temperature, wind, pressure). We assume the communication window is established up to the zenith angle \(\zeta_{max}=70^{\circ}\), when the difference between actual and perceived slant ranges is small [55]. While such limitation reduces the communication window and the size of data block, it also avoids contributions from the longest propagation distance overall and through the thickest air mass [56; 57]. Duration of communication window, i.e., total time in view of the satellite \(t\), determines the amount of accumulated data points \(N\) for given source repetition rate. The time in view is calculated from geometrical considerations and satellite orbital velocity, which for circular orbit in the observers meridian plane can be simplified as [58; 59]:  
\[t\approx 2\frac{(R_{\oplus}+H)^{3/2}}{\sqrt{G\cdot M_{\oplus}}}\left(\zeta_{ max}-\arcsin\left[\frac{R_{\oplus}}{R_{\oplus}+H}\sin\zeta_{max}\right]\right), \tag{6}\]  
where \(G\) is the gravitational constant, and \(M_{\oplus}\) is Earth mass.