  We show how weak non-linearities can be used in a device-independent quantum
key distribution (QKD) protocol using generalized two-mode Schr\"odinger cat
states. The QKD protocol is therefore shown to be secure against collective
attacks and for some coherent attacks. We derive analytical formulas for the
optimal values of the Bell parameter, the quantum bit error rate, and the
device-independent secret key rate in the noiseless lossy bosonic channel.
Additionally, we give the filters and measurements which achieve these optimal
values. We find that over any distance in this channel the quantum bit error
rate is identically zero, in principle, and the states in the protocol are
always able to violate a Bell inequality. The protocol is found to be superior
in some regimes to a device-independent QKD protocol based on polarization
entangled states in a depolarizing channel. Finally, we propose an
implementation for the optimal filters and measurements.
