  The core of security proofs of quantum key distribution (QKD) is the
estimation of a parameter that determines the amount of privacy amplification
that the users need to apply in order to distill a secret key. To estimate this
parameter using the observed data, one needs to apply concentration
inequalities, such as random sampling theory or Azuma's inequality. The latter
can be straightforwardly employed in a wider class of QKD protocols, including
those that do not rely on mutually unbiased encoding bases, such as the
loss-tolerant (LT) protocol. However, when applied to real-life finite-length
QKD experiments, Azuma's inequality typically results in substantially lower
secret-key rates. Here, we propose an alternative security analysis of the LT
protocol against general attacks, for both its prepare-and-measure and
measure-device-independent versions, that is based on random sampling theory.
Consequently, our security proof provides considerably higher secret-key rates
than the previous finite-key analysis based on Azuma's inequality. This work
opens up the possibility of using random sampling theory to provide alternative
security proofs for other QKD protocols.
