  Coherent state photon sources are widely used in quantum information
processing. In many applications, such as quantum key distribution (QKD), a
coherent state is functioned as a mixture of Fock states by assuming its phase
is continuously randomized. In practice, such a crucial assumption is often not
satisfied and, therefore, the security of existing QKD experiments is not
guaranteed. To bridge this gap, we provide a rigorous security proof of QKD
with discrete-phase-randomized coherent state sources. Our results show that
the performance of the discrete-phase randomization case is close to its
continuous counterpart with only a small number (say, 10) of discrete phases.
Comparing to the conventional continuous phase randomization case, where an
infinite amount of random bits are required, our result shows that only a small
amount (say, 4 bits) of randomness is needed.
