  The round-robin differential phase shift (RRDPS) quantum key distribution
(QKD) protocol is a unique quantum key distribution protocol whose security has
not been understood through an information-disturbance trade-off relation, and
a sufficient amount of privacy amplification was given independently of signal
disturbance. Here, we discuss the security of the RRDPS protocol in the
asymptotic regime when a good estimate of the bit error rate is available as a
measure of signal disturbance. The uniqueness of the RRDPS protocol shows up as
a peculiar form of information-disturbance trade-off curve. When the length of
a block of pulses used for encoding and the signal disturbance are both small,
it provides a significantly better key rate than that from the original
security proof. On the other hand, when the block length is large, the use of
the signal disturbance makes little improvement in the key rate. Our analysis
will bridge a gap between the RRDPS protocol and the conventional QKD
protocols.
