  We develop a general theory for quantum key distribution (QKD) in both the
forward error correction and the reverse error correction cases when the QKD
system is equipped with phase-randomized coherent light with arbitrary number
of decoy intensities. For this purpose, generalizing Wang's expansion, we
derive a convex expansion of the phase-randomized coherent state. We also
numerically check that the asymptotic key generation rates are almost saturated
when the number of decoy intensities is three.
