  We present methods to strictly calculate the finite-key effects in quantum
key distribution (QKD) with error rejection through two-way classical
communication (TWCC) for the sending-or-not-sending twin-field protocol. Unlike
the normal QKD without TWCC, here the probability of tagging or untagging for
each two-bit random group is not independent. We rigorously solve this problem
by imagining a virtual set of bits where every bit is independent and
identical. We show the relationship between the outcome starting from this
imagined set containing independent and identical bits and the outcome starting
with the real set of non-independent bits. With explicit formulas, we show that
simply applying Chernoff bound in the calculation gives correct key rate, but
the failure probability changes a little bit.

Quantum Physics