  We give a security proof of the `Round Robin Differential Phase Shift'
Quantum Key Distribution scheme, and we give a tight bound on the required
amount of privacy amplification. Our proof consists of the following steps. We
construct an EPR variant of the scheme. We identify Eve's optimal way of
coupling an ancilla to an EPR qudit pair under the constraint that the bit
error rate between Alice and Bob should not exceed a value beta. As a function
of beta we derive, for finite key size, the trace distance between the real
state and a state in which no leakage exists. For asymptotic key size we obtain
a bound on the trace distance by computing the von Neumann entropy. Our
asymptotic result for the privacy amplification is sharper than existing
bounds.
