  Many quantum key distribution (QKD) protocols require random choice of
measurement basis for each pulse or each train of pulses. In some QKD
protocols, such as the Round-Robin Differential Phase Shift (RRDPS) QKD
protocol, this requirement is a bit challenging as randomly choosing hundreds
of settings for every, say, 100 pulses may be too fast with current
technologies. In this paper, we solve this issue by proving the security of QKD
protocols with slow basis choice without compromising the secret key rate. We
also show that the random choice of the bases for the state preparation can be
made slow if the signals do not leak any information on the basis. Examples of
QKD protocols that our technique can apply include the RRDPS protocol and
BB84-type protocols, and our technique relaxes demands for the implementation
of QKD systems.
