  Device-independent quantum key distribution (DI-QKD) offers the strongest
form of security against eavesdroppers bounded by the laws of quantum
mechanics. However, a practical implementation is still pending due to the
requirement of combinations of visibility and detection efficiency that are
beyond those possible with current technology. This mismatch motivates the
search for DI-QKD protocols that can close the gap between theoretical and
practical security. In this work, we present two DI-QKD protocols whose
security relies on Bell inequalities with more than two inputs and two outputs.
We show that, for maximally entangled states and perfect visibility, a protocol
based on a Bell inequality with three inputs and four outputs requires a
slightly lower detection efficiency than the protocols based on Bell
inequalities with two inputs and two outputs.
