Lower bounds for the security of modified coherent-one-way quantum key distribution against one-pulse-attack

Upper bounds for the security of coherent-one-way (COW) quantum key distribution protocols have been analyzed by considering the one-pulse-attack [Branciard C, Gisin N and Scarani V (BGS) New J.Phys. (2008) 10 013031]. However, their security analysis was based on long distance case, and the typical value of the transmission distance is larger than 50 km. Applying the sharp continuity for the von Neumann entropy and some basic inequalities, we provide lower bounds for the security of modified coherent-one-way quantum key distribution protocol against the most general one-pulse-attack by only considering photon number resolved detectors that will be used in the receiver's side. Comparing with BGS's security analysis, our security analysis can be satisfied with arbitrary distance case.