  Device-independent quantum key distribution (DIQKD) allows two users to set
up shared cryptographic key without the need to trust the quantum devices used.
Doing so requires nonlocal correlations between the users. However, in [Phys.
Rev. Lett. 127, 050503 (2021)] it was shown that for known protocols
nonlocality is not always sufficient, leading to the question of whether there
is a fundamental lower bound on the minimum amount of nonlocality needed for
any DIQKD implementation. Here we show that no such bound exists, giving
schemes that achieve key with correlations arbitrarily close to the local set.
Furthermore, some of our constructions achieve the maximum of 1 bit of key per
pair of entangled qubits. We achieve this by studying a family of
Bell-inequalities that constitute all self-tests of the maximally entangled
state with a single linear Bell expression. Within this family there exist
non-local correlations with the property that one pair of inputs yield outputs
arbitrarily close to perfect key. Such correlations exist for a range of
Clauser-Horne-Shimony-Holt (CHSH) values, including those arbitrarily close to
the classical bound. Finally, we show the existence of quantum correlations
that can generate both perfect key and perfect randomness simultaneously,
whilst also displaying arbitrarily small CHSH violation; this opens up the
possibility of a new class of cryptographic protocol.
