  Device-independent (DI) protocols, such as DI conference key agreement
(DICKA) and DI randomness expansion (DIRE), certify private randomness by
observing nonlocal correlations when two or more parties test a Bell
inequality. While most DI protocols are restricted to bipartite Bell tests,
harnessing multipartite nonlocal correlations may lead to better performance.
Here, we consider tripartite DICKA and DIRE protocols based on testing
multipartite Bell inequalities, specifically: the
Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and the Holz and the
Parity-CHSH inequalities introduced in the context of DICKA protocols. We
evaluate the asymptotic performance of the DICKA (DIRE) protocols in terms of
their conference key rate (net randomness generation rate), by deriving lower
bounds on the conditional von Neumann entropy of one party's outcome and two
parties' outcomes. For the Holz inequality, we prove a tight analytical lower
bound on the one-outcome entropy and conjecture a tight lower bound on the
two-outcome entropy. We additionally re-derive the analytical one-outcome
entropy bound for the MABK inequality with a much simpler method and obtain a
numerical lower bound on the two-outcome entropy for the Parity-CHSH
inequality. Our simulations show that DICKA and DIRE protocols employing
tripartite Bell inequalities can significantly outperform their bipartite
counterparts. Moreover, we establish that genuine multipartite entanglement is
not a precondition for multipartite DIRE while its necessity for DICKA remains
an open question.
