## 1 Introduction  
Quantum key distribution (QKD) [1, 2, 3] is well known to have its goal in developing methods (protocols) for sharing a secret key between legitimate users, who can lately use the key for the confidential information transfers. First started with the discrete-variable protocols based on direct detection of single-photon states (and their emulation using weak coherent pulses or entangled photon pairs [4]), QKD was later extended to the realm of continuous variables (CV) [5] based on efficient and low-noise homodyne detection of multiphoton coherent or squeezed states of light.  
One of the important applications of QKD is in the extra-terrestrial channels, which potentially allow extremely long-distance secure communication enabled by QKD over a satellite. While discrete-variable protocols were recently successfully tested over the satellite links [6, 7, 8, 9], the applicability of satellite-based CV QKD remains less studied. Indeed, it was considered in the asymptotic regime of the infinitely many quantum states [10, 11], which however is never the case in practice. Moreover, CV QKD may have an important practical advantage in free-space applications and particularly in the satellite-based channels because a homodyne detector, in which the signal is coupled to a narrow-band local oscillator (bright coherent beam used as a phase reference), intrinsically filters out the background radiation at unmatched wavelengths [12]. Thus CV QKD can operate in conditions of strong stray light and potentially at daytime, which, in the case of discrete-variableprotocols, would require additional filtering, increasing attenuation and complexity of the set-up. Recently, the feasibility of coherent-state CV QKD over satellite links was discussed in [13]. In the current work we analyze applicability of CV QKD over satellite-based channels considering also squeezed signal states. As the feasible squeezing up 10 dB, achievable with current technology [14], is known to improve robustness of CV QKD to noise [15; 16; 17], we confirm its usefulness in the satellite-based links as well. We build the channel model on the assumption of normal fluctuation of deflected signal beam center around the receiving aperture center and study applicability of CV QKD, taking into account the finite data ensemble size. We show that in this regime the protocols appear to be extremely sensitive to strong channel attenuation and large amounts of data are required for successful realization of CV QKD over satellites, which contradicts relatively short passage times. Possible solutions to circumvent the problem can be (i) use of squeezed states, that reduce requirements on the data ensemble size and can tolerate stronger attenuation and channel noise; (ii) relaxation of security assumptions considering individual attacks or passive eavesdropping, introducing no excess noise; (iii) increase of the link transmittance using larger telescopes in the downlink regime; (iv) increase of the repetition rate of the system in order to accumulate larger statistics. Our results reveal substantial challenges for implementing CV QKD over satellites but shows no fundamental limits for such realizations. Already under the strict assumption of collective eavesdropping attacks and untrusted channel noise, CV QKD protocols using feasible squeezing should be applicable with low-orbit satellites, while in the assumption of passive eavesdropping squeezed-state CV QKD can tolerate up to 43 dB of channel attenuation, which paves the way towards realization over geostationary satellites.