  Recent studies showed the finite-size security of binary-modulation CV-QKD
protocols against general attacks. However, they gave poor key-rate scaling
against transmission distance. Here, we extend the security proof based on
complementarity, which is used in the discrete-variable QKD, to the previously
developed binary-modulation CV-QKD protocols with the reverse reconciliation
under the finite-size regime and obtain large improvements in the key rates.
Notably, the key rate in the asymptotic limit scales linearly against the
attenuation rate, which is known to be optimal scaling but is not achieved in
previous finite-size analyses. This refined security approach may offer
full-fledged security proofs for other discrete-modulation CV-QKD protocols.
