  We show the information-theoretic security proof of the
differential-phase-shift (DPS) quantum key distribution (QKD) protocol based on
the complementarity approach [arXiv:0704.3661 (2007)]. Our security proof
provides a slightly better key generation rate compared to the one derived in
the previous security proof in [arXiv:1208.1995 (2012)] that is based on the
Shor-Preskill approach [Phys. Rev. Lett. ${\bf 85}$, 441 (2000)]. This
improvement is obtained because the complementarity approach can employ more
detailed information on Alice's sending state in estimating the leaked
information to an eavesdropper. Moreover, we remove the necessity of the
numerical calculation that was needed in the previous analysis to estimate the
leaked information. This leads to an advantage that our security proof enables
us to evaluate the security of the DPS protocol with any block size. This paper
highlights one of the fundamental differences between the Shor-Preskill and the
complementarity approaches.
