We consider an optimal quantum key distribution setup based on minimal number of measurement bases with binary yields used by parties against an eavesdropper limited only by the no-signaling principle. We note that in general, the maximal key rate can be achieved by determining the optimal tradeoff between measurements that attain the maximal Bell violation and those that maximise the bit correlation between the parties. We show that higher correlation between shared raw keys at the expense of maximal Bell violation provide for better key rates for low channel disturbance.
We consider the counterfactual protocol proposed in Phys. Rev. Lett., 103, 230501 (2009) within a device independent framework and show how its security can easily be compromised. Capitalising on the fact that the protocol is based on the use of a single photon entanglement phenomenon, we propose an equivalent protocol. It can be made secure within such a pessimistic framework against a supra-quantum Eve limited only by the no-signalling principle. The equivalence the protocol demonstrates the possibility of device independent framework for counterfactual quantum cryptography.