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We develop a general theory for quantum key distribution (QKD) in both theforward error correction and the reverse error correction cases when the QKDsystem is equipped with phase-randomized coherent light with arbitrary numberof decoy intensities. For this purpose, generalizing Wang's expansion, wederive a convex expansion of the phase-randomized coherent state. We alsonumerically check that the asymptotic key generation rates are almost saturatedwhen the number of decoy intensities is three.Comment: This manuscript has been revised extensivel

General theory for decoy-state quantum key distribution with an arbitrary number of intensities