BDSW protocol revisited: an efficient method for the key distillation without classical computational complexity

In quantum key distribution(QKD), one can use a classical CSS code to distill the final key. However, there is a constraint for the two codes in CSS code and so far it is unknown how to construct a large CSS code efficiently. Here we show that the BDSW method given by Bennett et al can be modified and the error correction and privacy amplification can be done separately with two INDEPENDENT parity matrices. With such a modification, BDSW method can be used to distill the final key without any classical computational complexity. We also apply the method to the case of imperfect source where a small fraction of signals are tagged by Eve.