## 4 Security Analysis in Composable Finite-Size Regime
### With Clusterization  
In this subsection, we discuss the performance of the clusterization method. Figure 5 shows the composable finite-size rate \(R_{\text{com}}\) with clusterization versus the cluster number, \(C\), and the intervals. The satellite altitudes of the downlink and uplink scenarios are 300 km and 100 km, respectively. Figure 5a shows the rate of the downlink scenario versus the interval with various cluster numbers. Here, we also provide the corresponding asymptotic key rate for comparison, where \(C=1\) denotes the case without clusterization. We find that the clusterization method can improve the composable finite-size rate after the seventh interval, even with only two clusters. Moreover, the composable finite-size rates of \(C=2\) and \(C=10\) surpass the asymptotic key rate after the 12 th and 11 th intervals, respectively. This is because a bigger interval number means a larger zenith angle and thus a higher \(\bar{n}_{\text{fad}}\) (see Table 3), while the proposed clusterization method can suppress it. In other words, more cluster numbers lead to less fading noise. However, having more cluster numbers means a more serious composable finite-size effect. Take \(C=10\) as an example: the clusterization method damages the rate before the sixth interval, where \(\bar{n}_{\text{fad}}\) is at a relatively low level. In these intervals, the added composable finite-size effect becomes the main effect, and the improvement caused by clusterization is overshadowed by this effect. When the fading noise is large enough, the rate of the clusterization method is consistently higher than that of no clusterization, as shown in Figure 5b and Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Notation** & **Value** \\ \hline Digitalization & \(d\) & \(2^{5}\) \\ Hashing parameter & \(\varepsilon_{\text{h}}\) & \(10^{-10}\) \\ Smoothing parameter & \(\varepsilon_{\text{s}}\) & \(10^{-10}\) \\ \(\varepsilon\)-correctness & \(\varepsilon_{\text{cor}}\) & \(10^{-10}\) \\ Failing probability of PE 1 & \(\varepsilon_{\text{pe}}\) & \(10^{-10}\) \\ EC 2 success probability & \(\mathcal{P}_{\text{ec}}\) & \(0.9\) \\ Proportion of PE data & \(r\) & \(0.9\) \\ Frequency of signal preparation and detection & \(v_{s}\) & \(1\,\text{GHz}\), \(10\,\text{GHz}\) \\ Frequency of channel fluctuation & \(v_{f}\) & \(1\,\text{KHz}\) \\ Total pulses (one minute) & \(N\) & \(60\cdot v_{s}\) \\ Individual package number & \(m\) & \(N/(v_{s}\cdot v_{f})\) \\ Pulses of an individual package & \(n\) & \(N/m\) \\ Cluster number & \(C\) & \(1\)–\(10\) \\ \hline \hline \end{tabular} \({}^{\dagger}\) PE denotes parameter estimation. 2 EC denotes error correction.  
\end{table} 
Table 2: Parameters related to composable finite-size security. Figure 4: The composable finite-size rate with no clusterization versus the satellite altitude and zenith angle: (**a**) downlink scenario, (**b**) uplink scenario. The color bars are the same as those in Figure 2.  
Figure 5c shows the downlink rate versus the cluster number at specific intervals. Here, we also provide the results of \(v_{s}=10\) GHz for comparison. In the 1 GHz case, we find that increasing the number of clusters reduces the rate in the fifth interval since its fading noise is relatively low. For the 10 th interval, there is an optimal cluster number of \(C_{\rm opt}=6\). Beyond this number, the increase in clusters cannot further improve performance since the composable finite-size effect increases. However, the results show a difference when \(v_{s}\) increases from 1 GHz to 10 GHz. In the 10 GHz case, the composable finite-size effect is also visible in the 1 GHz case. Figure 5: The composable finite-size rate with clusterization versus the cluster number and interval. (**a,c**) are downlink scenarios with a satellite altitude of \(h=300\) km, (**b,d**) are uplink scenarios with a satellite altitude of \(h=100\) km. The value of \(n_{\rm fad}\) in each interval refers to the information in Table 3.