Composable security for practical quantum key distribution with two way classical communication

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 sending-or-not-sending twin-field protocol. Unlike normal QKD without TWCC, here probability of tagging or untagging each two-bit random group is not independent. rigorously solve this problem by imagining a virtual set bits where every bit independent and identical. show relationship between outcome starting from imagined containing identical real non-independent bits. With explicit formulas, we that simply applying Chernoff bound calculation gives correct rate, but failure changes little bit.