  Device-independent quantum key distribution aims at providing security
guarantees even when using largely uncharacterised devices. In the simplest
scenario, these guarantees are derived from the CHSH score, which is a simple
linear combination of four correlation functions. We here derive a security
proof from a generalisation of the CHSH score, which effectively takes into
account the individual values of two correlation functions. We show that this
additional information, which is anyway available in practice, allows one to
get higher key rates than with the CHSH score. We discuss the potential
advantage of this technique for realistic photonic implementations of
device-independent quantum key distribution.
