  We study the sending-or-not-sending (SNS) protocol with discrete phase
modulation of coherent states. We first make the security of the SNS protocol
with discrete phase modulation. We then present analytic formulas for key rate
calculation. We take numerical simulations for the key rate through discrete
phase modulation of both the original SNS protocol and the SNS protocol with
two way classical communications of active-odd-parity pairing (AOPP). Our
numerical simulation results show that only with $6$ phase values, the key
rates of the SNS protocol can exceed the linear bound, and with $12$ phase
values, the key rates are very close to the results of the SNS protocol with
continuously modulated phase-randomization.
