  We provide a simple security proof for prepare & measure quantum key
distribution protocols employing noisy processing and one-way postprocessing of
the key. This is achieved by showing that the security of such a protocol is
equivalent to that of an associated key distribution protocol in which, instead
of the usual maximally-entangled states, a more general {\em private state} is
distilled. Besides a more general target state, the usual entanglement
distillation tools are employed (in particular, Calderbank-Shor-Steane
(CSS)-like codes), with the crucial difference that noisy processing allows
some phase errors to be left uncorrected without compromising the privacy of
the key.
