We provide an analysis of a new family of device independent quantum key distribution (QKD) protocols with several novel features: (a) The bits used for the secret key do not come from the results of the measurements on an entangled state but from the choices of settings; (b) Instead of a single security parameter (a violation of some Bell inequality) a set of them is used to estimate the level of trust in the secrecy of the key. The main advantage of these protocols is a smaller vulnerability to imperfect random number generators made possible by feature (a). We prove the security and the robustness of such protocols. We show that using our method it is possible to construct a QKD protocol which retains its security even if the source of randomness used by communicating parties is strongly biased. As a proof of principle, an explicit example of a protocol based on the Hardy's paradox is presented. Moreover, in the noiseless case, the protocol is secure in a natural way against any type of memory attack, and thus allows to reuse the device in subsequent rounds. We also analyse the robustness of the protocol using semi-definite programming methods. Finally, we present a post-processing method, and observe a paradoxical property that rejecting some random part of the private data can increase the key rate of the protocol.