Differential-phase-shift quantum key distribution protocol with small number of random delays

The differential-phase-shift (DPS) quantum key distribution (QKD) protocol was proposed aiming at simple implementation, but it can tolerate only a small disturbance in a quantum channel. The round-robin DPS (RRDPS) protocol could be a good solution for this problem, which in fact can tolerate even up to $50\%$ of a bit error rate. Unfortunately, however, such a high tolerance can be achieved only when we compromise the simplicity, i.e., Bob's measurement must involve a large number of random delays ($|\mathcal{R}|$ denotes its number), and in a practical regime of $|\mathcal{R}|$ being small, the tolerance is low. In this paper, we propose a new DPS protocol to achieve a higher tolerance than the one in the original DPS protocol, in which the measurement setup is less demanding than the one of the RRDPS protocol for the high tolerance regime. We call the new protocol the small-number-random DPS (SNRDPS) protocol, and in this protocol, we add only a small amount of randomness to the original DPS protocol, i.e., $2\leq|\mathcal{R}|\leq10$. In fact, we found that the performance of the SNRDPS protocol is significantly enhanced over the original DPS protocol only by employing a few additional delays such as $|\mathcal{R}|=2$. Also, we found that the key generation rate of the SNRDPS protocol outperforms the RRDPS protocol without monitoring the bit error rate when it is less than $5\%$ and $|\mathcal{R}|\leq10$. Our protocol is an intermediate protocol between the original DPS protocol and the RRDPS protocol, and it increases the variety of the DPS-type protocols with quantified security.