## 2 System Description
\right], \tag{3}\]  
where \(\langle\tau\rangle=\sum_{j=1}^{K}\mathcal{P}^{j}\tau^{j}\), \(\langle\sqrt{\tau}\rangle=\Sigma_{j=1}^{K}\mathcal{P}^{j}\sqrt{\tau}\), and \(\bar{n}\) is the total number of thermal photons due to the various sources of noise. Note that the above covariance matrix takes the detector efficiency \(\eta_{\mathrm{eff}}\) (considered constant) into account, i.e., \(\tau^{j}=\eta_{\mathrm{eff}}\eta_{\mathrm{ch}}^{j}\). Finally, [PERSON] performs shot-noise limited homodyne detection (or heterodyne detection) on mode \(B_{1}\) and shares secret keys with [PERSON] through classical data post-processing (i.e., parameter estimation, error correction, and privacy amplification). Next, we derive the asymptotic key rate of the protocol under Gaussian collective attacks. Here, we use reverse reconciliation (RR) in data post-processing, which is more robust against channel attenuation compared to direct reconciliation [26]. In terms of the Csiszar-Korner theorem, the asymptotic key rate under RR is given by [31; 33]  
\[R_{\rm asy}\big{(}\langle\tau\rangle,\langle\sqrt{\tau}\rangle,\bar{n}\big{)}= \beta I_{AB}-\chi_{BE}, \tag{4}\]  
where \(\beta\) denotes the reconciliation efficiency, \(I_{AB}\) is the Shannon mutual information between [PERSON] and [PERSON], and \(\chi_{BE}\) represents the Holevo bound. The values of \(I_{AB}\) and \(\chi_{BE}\) can be derived from the covariance matrix in Equation (3). Therefore, we first need to know the values of \(\langle\tau\rangle\), \(\langle\sqrt{\tau}\rangle\), and \(\bar{n}\). In our manuscript, we use the model in [23] to describe the PDT of satellite-mediated links, which takes the diffraction, atmospheric extinction, turbulence, and pointing errors into account. Moreover, \(\bar{n}\) is considered to originate from background light, i.e., \(\bar{n}=\eta_{\rm eff}\bar{n}_{B}\), while the setup noises on both [PERSON]'s and [PERSON]'s side are ignored (see [29] for a discussion of setup noise). Once we determine \(\langle\tau\rangle\), \(\langle\sqrt{\tau}\rangle\), and \(\bar{n}\), the derivation of \(I_{AB}\) and \(\chi_{BE}\) can be found (refer to [22]). Note that our manuscript considers [PERSON]'s active attack on the channel to establish a lower bound on the secret key rate. However, the satellite-mediated link can be treated as a line-of-sight (LOS) link, as [PERSON] would find it hard to tamper with the middle channel between the ground station and the moving satellite, and would likely only conduct passive attacks. The LOS security is discussed in [30], where it is shown that LOS scenarios can achieve higher secret key rates than cases involving active attacks. Figure 2 shows the asymptotic key rate, \(R_{\rm asy}\), of the GMCS protocol over the satellite-mediated links versus the satellite altitude \(h\) and the zenith angle \(\theta\). The simulation parameters are shown in Table 1. Here, we consider the nighttime situation with an atmospheric parameter of \(A_{0}=1.7\times 10^{-14}\) m\({}^{-2/3}\) and a windspeed of \(v=21\) m/s, which is commonly known as  
Figure 1: (**a**) EB scheme of GMCS protocol over a satellite-mediated (fading) link. Note that [PERSON] can also use heterodyne detection instead of homodyne detection. (**b**) Demonstration of the clusterization method. EPR: Einstein–Podolsky–Rosen; PDT: probability distribution of transmittance; \(\eta_{\rm ch}\): channel transmittance; \(\bar{n}_{B}\): mean number of background thermal photons; \(C\): cluster number; \(\tau\): real-time transmittance including detector efficiency; \(\tau_{\rm max}^{k}\) (\(\tau_{\rm min}^{k}\)): the upper (lower) bound of the \(k\)-th cluster, \(k=1,\dots,C\). the H-V\({}_{5/7}\) model. Moreover, \(\bar{n}_{\rm B}\) is set to 0.02 for both the downlink and uplink scenarios, as our focus is primarily on the effects caused by channel fading instead of background noise. In fact, the background noise shows a difference between the downlink and uplink scenarios due to variations in the source of background light [23].