Using Cascade in quantum key distribution

We point out a critical flaw in the analysis of quantum key distribution protocols that employ the twoall two-way communication that occurs during the Cascade protocol. We present a straightforward and elegant alternative approach that addresses this flaw and produces valid key rates. We exemplify our new approach by comparing its key rates with those generated using older, incorrect approaches, for qubit BB84 and decoy-state BB84 protocols. We show that, in many practically relevant situations, our rectified approach produces the same key rate as older, incorrect approaches. However, in other scenarios, our approach produces valid key rates that are lower, highlighting the importance of properly accounting for all two-way communication during Cascade.