  Differential phase shift quantum key distribution systems have a high
potential for achieving high speed key generation. However, its unconditional
security proof is still missing, even though it has been proposed for many
years. Here, we prove its security against collective attacks with a weak
coherent light source in the noiseless case (i.e. no bit error). The only
assumptions are that quantum theory is correct, the devices are perfect and
trusted and the key size is infinite. Our proof works on threshold detectors.
We compute the lower bound of the secret key generation rate using the
information-theoretical security proof method. Our final result shows that the
lower bound of the secret key generation rate per pulse is linearly
proportional to the channel transmission probability if Bob's detection counts
obey the binomial distribution.
