## 3 Satellite-to-Ground QKD Simulator
### Protocol Assumptions and Finite Key Size Analysis  
Since the atmospheric polarization decoherence of the atmosphere has been measured as being limited [73], polarization-based QKD protocols appear to be suitable for satellite-to-ground QKD communication. For this reason, the efficient Decoy-State BB84 (DS-BB84) protocol, which employs the \"weak + vacuum\" configuration detailed in [74], was selected. The protocol integrates decoy and vacuum states into the traditional BB84 framework to mitigate the Photon Number Splitting (PNS) attacks. This incorporation enables precise assessments of background noise and channel loss, enhancing the protocol's ability to detect eavesdropping attempts. Consequently, the decoy state protocol exhibits decreased susceptibility to PNS attacks, thus allowing for higher mean photon numbers per pulse. The finite key size effect was considered for the protocol modeling. Finite key size in QKD refers to the limited number of secret key bits that can be extracted from the quantum signals exchanged between communicating parties. It is a critical consideration in QKD protocols because a finite key size directly affects the security of the shared key. With a limited number of key bits, the statistical fluctuations between ones and zeros should be considered as these result in disruptions of the key homogeneity, which increase information leakage or induce key compromises through specific attacks (such as collective attacks) [75; 76]. Therefore, accounting for finite key size is essential to accurately assess the security level of the key generated through QKD, as well as to implement appropriate error correction and privacy amplification techniques to mitigate the effects of finite-size effects and ensure the integrity of the secret key. When considering a finite post processing block size, the statistical fluctuations of the visibility were included. \(Q_{\mu}\) corresponds to the gain of the signal state and \(E_{\mu}\) corresponds to the overall QBER. The values of \(Q_{\mu}\) and \(E_{\mu}\) can be calculated as  
\[Q_{\mu}=Y_{0}+exp(-\eta\mu), \tag{16}\]  
\[E_{\mu}=\frac{e_{0}Y_{0}+e_{sys}(1-exp(-\eta\mu))}{Q_{\mu}}, \tag{17}\]  
where \(\eta\) corresponds to the overall link transmittance; \(e_{sys}\) corresponds to the baseline system error rate and is equal to \((1-\mathrm{V})/2\) (where V corresponds to the total polarization visibility of the Bob station, i.e., the limited Polarization Extinction Ratio (PER) of the Bob's polarization analysis module, as well the polarization error due to polarization basis deviation); \(e_{0}\) corresponds to the error rate of the background noise; \(e_{0}\) is set to \(1/2\) by assuming that the background noise is random; and \(Y_{0}\) corresponds to the probability of the detector to click due to dark counts, background noise, or due to after pulsing probability (for the case of SPAD detection). The latter is expressed as follows:  
\[Y_{0}=P_{dc}+P_{back}+P_{ap}. \tag{18}\]\(P_{back}\) can be calculated according to the background radiance value as follows:  
\[P_{back}=t_{gate}\ \eta_{Rx}\ \eta_{FC}\frac{P_{back}}{h\cdot f}, \tag{19}\]  
where \(h\cdot f\) correspond to the single-photon energy; \(t_{gate}\) to the effective detection time window; \(\eta_{Rx}\) and \(\eta_{FC}\) to the receiver system (filter loss, polarization controller loss, etc. ), and fiber coupling overall efficiencies, respectively; and \(P_{back}\) to the receiver background solar radiance, which is provided by Equation (19). For the calculation of the SKR, we assumed security against restricted collective attacks and followed the methodology reported in [75; 76].