  We prove the security of the Bennett-Brassard (BB84) quantum key distribution
protocol in the case where the key information is encoded in the relative phase
of a coherent-state reference pulse and a weak coherent-state signal pulse, as
in some practical implementations of the protocol. In contrast to previous
work, our proof applies even if the eavesdropper knows the phase of the
reference pulse, provided that this phase is not modulated by the source, and
even if the reference pulse is bright. The proof also applies to the case where
the key is encoded in the photon polarization of a weak coherent-state pulse
with a known phase, but only if the phases of the four BB84 signal states are
judiciously chosen. The achievable key generation rate scales quadratically
with the transmission in the channel, just as for BB84 with phase-randomized
weak coherent-state signals (when decoy states are not used). For the case
where the phase of the reference pulse is strongly modulated by the source, we
exhibit an explicit attack that allows the eavesdropper to learn every key bit
in a parameter regime where a protocol using phase-randomized signals is
provably secure.
