  Multiparty quantum cryptography based on distributed entanglement will find
its natural application in the upcoming quantum networks. The security of many
multipartite device-independent (DI) protocols, such as DI conference key
agreement, relies on bounding the von Neumann entropy of the parties' outcomes
conditioned on the eavesdropper's information, given the violation of a
multipartite Bell inequality. We consider three parties testing the
Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of
their outcomes by bounding the conditional entropy of a single party's outcome
and the joint conditional entropy of two parties' outcomes. From the former
bound, we show that genuine multipartite entanglement is necessary to certify
the privacy of a party's outcome, while the latter significantly improve
previous results. We obtain the entropy bounds thanks to two general results of
independent interest. The first one drastically simplifies the quantum setup of
an $N$-partite Bell scenario. The second one provides an upper bound on the
violation of the MABK inequality by an arbitrary $N$-qubit state, as a function
of the state's parameters.
