## 3 CV QKD over Satellite Channels
### Security Evaluation
Data clusterization with respect to channel attenuation allows to compensate the effect of channel transmittance fluctuations [53]; however, such postprocessing can be demanding for satellite-based QKD and is not needed for slow systematic changes of transmittance during the satellite pass. In the current work we therefore adopt simpler albeit similar method by splitting the satellite tracked pass into a set of segments and generating the key for each one. This allows for the achievement of the following lower bound on the overall secure key rate as a weighed sum of the key rates from individual segments:  
\[K=\sum_{i}\max\Bigl{\{}0,\frac{n_{i}}{N}[\beta I_{AB}-\chi_{BE}-\delta(n_{i})] \Bigr{\}}, \tag{13}\]  
where \(n_{i}\) is the raw key length for a given segment \(i\) with number of data points \(N_{i}\), so that \(\sum_{i}N_{i}=N\), \(N_{i}-n_{i}\) of data points is used for segment channel estimation, and the weight is determined by the relative size of the segment \(n_{i}/N\), with \(N\) being the overall block size for  
Figure 4: (**Left**) Asymptotic lower bound on the key rate (in bits per channel use) of the optimized coherent-state (dashed) and squeezed-state CV QKD protocols (solid) secure against active individual attacks for a single pass of a LEO satellite with orbit altitude \(H\) and receiver aperture radius \(a=0.5\), 0.75 m (green and blue lines respectively). Channel excess noise at the output \(\epsilon=10^{-4}\) SNU. (**Right**) The key rate (in bits per channel use) of optimized coherent-state (dashed) and squeezed-state protocol (solid) under passive collective attacks for a single pass of a LEO satellite with orbit altitude \(H\), and receiver aperture radius \(a=0.5\), 0.75 m (green and blue lines respectively). Note that the coherent-state protocol can only be securely established with larger aperture \(a=0.75\) m. The overall block size depends on the length of communication window (given by the orbit altitude \(h\)) and the clock rate (from bottom to top) 100 MHz, 1 GHz and 10 GHz. Trusted channel noise at the output is \(10^{-4}\) SNU. Both squeezing and modulation variance are optimized (with optimal squeezing limited by a feasible value as \(V_{S}\geq 0.1\)) in order to establish a secure key regardless of the altitude. a given satellite pass. The segments are chosen in accordance with the zenith angle \(\zeta\), so that for \(i=1,2,3\) we obtain three segments each containing measurement results at respectively \([-\zeta_{max},-2/3\,\zeta_{max})\cup(2/3\,\zeta_{max},\zeta_{max}]\), \([-2/3\,\zeta_{max},-1/3\,\zeta_{max})\cup(1/3\,\zeta_{max},2/3\,\zeta_{max}]\), and \([-1/3\,\zeta_{max},1/3\,\zeta_{max}]\). The finite-size effects are stronger within each segment and transmittance fluctuations are substantial, yet reduced. Three segments are already sufficient to attain an enhanced positive secure key rate and extend the range of secure altitudes, as shown in Figure 6. For systems with smaller apertures this implies an effective increase of no tracking zone \(\zeta_{max}\), as the segment characterized by the longer slant ranges \(L(\zeta)\) might not contribute to the overall key.