Upper bounds of eavesdropper's performances in finite-length code with decoy method

Security formulas of quantum key distribution (QKD) with imperfect resources are obtained for finite-length code when the decoy method is applied. This analysis is useful for guaranteeing the security of implemented QKD systems. Our formulas take into account the effect of the vacuum state and dark counts in the detector. We compare the asymptotic key generation rate in presence of dark counts with that without.