  We point out a critical flaw in the analysis of Quantum Key Distribution
(QKD) protocols that employ the two-way error correction protocol Cascade.
Specifically, this flaw stems from an incom-plete consideration of all 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.
