  The Gaussian modulated continuous variable quantum key distribution
(GM-CVQKD) protocol is known to maximise the mutual information between two
parties during quantum key distribution (QKD). An alternative modulation scheme
is the discrete modulated CVQKD (DM-CVQKD) protocol. In this paper, we study
the Phase Shift Keying (M-PSK) and Quadrature Amplitude Modulation (M QAM)
DM-CVQKD protocols along with the GM-CVQKD protocol over a satellite-to-ground
link in the low SNR regime. We use a satellite-to-ground link model which takes
into account geometric losses, scintillation, and scattering losses from the
link distance, atmospheric turbulence, and atmospheric aerosols, respectively.
In addition, recent multidimensional (MD) and multilevel coding and multistage
decoding (MLC-MSD) reconciliation method models in combination with
multiedge-type low-density parity-check (MET-LDPC) code models have been used
to determine the reconciliation efficiency. The results show that GM-CVQKD
outperforms DM-CVQKD. In addition, GM-CVQKD with MD reconciliation outperforms
GM-CVQKD with MLC-MSD reconciliation in the finite size limit by producing
positive secret key rates at larger link distances and lower elevation angles.
