## 3 CV QKD over Satellite Channels
### Security Evaluation  
We first assess the security by looking at the impact of individual attacks in asymptotic regime (see Figure 4 left). In this regime, security can be established for every LEO satellite altitude and feasible squeezing \(V_{S}\geq 0.1\) (SNU) provides a noticeable rate gain. Note that under the individual attacks, the higher levels of squeezing always translate into a higher secure key rate, which is not the case for collective attacks where the transmittance fluctuations limit the applicable values of squeezing [66] and require active squeezing optimization based on the estimated atmospheric transmittance distribution. The optimization of the latter is a crucial step required for the establishing of secure key rate based on the data generated from a single pass of the satellite. The performance of the optimized CV QKD protocol under passive collective attacks with trusted noise \(\epsilon=10^{-4}\) is depicted in Figure 4 (right). The impact of finite-size effects reduces with an increase of the altitude \(H\) as the communication window gets longer and consequently the overall block size \(N\) gets larger. Low altitude satellite downlinks exhibit less mean attenuation but such advantage is partially offset by larger confidence intervals of estimated channel parameters and shorter raw key. While optimization of estimation block-size \(N-n\) can lengthen the key to some extent, increasing the repetition rate of the system is necessary to greatly extend the raw key. Higher repetition rate is especially crucial for the coherent-state protocol that can be implemented at the altitudes above 500 km only with 10 GHz clock rates. Increasing the size of receiving aperture also leads to significant improvement of the secure key rate. Mean channel attenuation and fading noise, originating from transmittance fluctuations, diminish the tolerance to the channel excess noise, as shown in Figure 5. Evidently,  
Figure 3: Mean signal loss in satellite-to-ground channel dependence on orbit altitude \(H\) with receiver aperture radius \(a=0.5\), 0.75, 1 m (from top to bottom respectively). Single pass transmittance distribution is assembled from simulated individual links at every zenith angle within \(-\zeta_{max}<\zeta<\zeta_{max}\). both protocols are extremely sensitive to excess noise and no secure key can be generated at any orbit altitude if the noise at the output of the squeezed-state protocol is \(\epsilon\geq 2\cdot 10^{-3}\) SNU for \(a=0.5\), or if the noise \(\epsilon\geq 5\cdot 10^{-3}\) for \(a=0.75\), with orders of magnitude lower values needed for security of the coherent-state protocol (\(\epsilon\geq 6\cdot 10^{-5}\) at \(a=0.5\), or \(\epsilon\geq 4\cdot 10^{-4}\) at \(a=0.75\)). Note that with an increase of satellite orbit altitude, the rate at which the protocol loses noise tolerance decreases. As the optical channel becomes longer it also becomes more stable [48; 51], i.e., both mean attenuation and fading (viewed as the variance of transmittance fluctuations \(\langle\eta_{tot}\rangle-\langle\sqrt{\eta_{tot}}\rangle^{2}\)) are simultaneously reduced. While the effect at LEO altitudes is more apparent for the coherent-state protocol, the same dependency can be expected for the squeezed-state protocol at higher altitudes (MEO or GEO). Furthermore, channel stabilization will be more pronounced with less accurate beam-tracking \(\theta_{p}\) which directly limits the beam-wandering and consequently channel fading. Clearly, in order to operate under collective attacks with untrusted channel noise, the noise has to be limited to very low values. This can be achieved by proper control of the set-up or precise parameter estimation; however, one can as well reduce the amount of fading noise by dividing the overall single-pass data block into a subset of smaller blocks [48].