  In general, generation of entangled photon pairs and also the random choice
of measurement basis can be implemented with passive optical devices in
entanglement based quantum key distribution (QKD) system. However, auxiliary
random numbers are still mandatory for conducting the privacy amplification
procedure. In this paper, we propose a fully passive entanglement based QKD
scheme in which certified random bits for privacy amplification are extracted
from detection events of measurement-basis-mismatched entangled photon pairs.
The privacy of these random bits can be certified by estimating the phase error
rate in the sifted key. The idea of extracting certified random numbers from
measurement-basis-mismatched photons, can be extended to all practical QKD
schemes.
