This is a study of the security of the Coherent One-Way (COW) protocol for
quantum cryptography, proposed recently as a simple and fast experimental
scheme. In the zero-error regime, the eavesdropper Eve can only take advantage
of the losses in the transmission. We consider new attacks, based on
unambiguous state discrimination, which perform better than the basic
beam-splitting attack, but which can be detected by a careful analysis of the
detection statistics. These results stress the importance of testing several
statistical parameters in order to achieve higher rates of secret bits
