  Discrete-Modulated (DM) Continuous-Variable Quantum Key Distribution (CV-QKD)
protocols are promising candidates for commercial implementations of quantum
communication networks due to their experimental simplicity. While tight
security analyses in the asymptotic limit exist, proofs in the finite-size
regime are still subject to active research. We present a composable
finite-size security proof against independently and identically distributed
collective attacks for a general DM CV-QKD protocol. We introduce a new energy
testing theorem to bound the effective dimension of Bob's system and rigorously
prove security within Renner's epsilon-security framework and address the issue
of acceptance sets in protocols and their security proof. We want to highlight,
that our method also allows for nonunique acceptance statistics, which is
necessary in practise. Finally, we extend and apply a numerical security proof
technique to calculate tight lower bounds on the secure key rate. To
demonstrate our method, we apply it to a quadrature phase-shift keying
protocol, both for untrusted, ideal and trusted nonideal detectors. The results
show that our security proof method yields secure finite-size key rates under
experimentally viable conditions up to at least 72km transmission distance.
