  Local measurements on bipartite maximally entangled states can yield
correlations that are maximally nonlocal, monogamous, and associated to fully
random outcomes. This makes these states ideal for bipartite cryptographic
tasks. Genuine-multipartite nonlocality constitutes a stronger notion of
nonlocality that appears in the multipartite case. Maximal genuine-multipartite
nonlocality, monogamy and full random outcomes are thus highly desired
properties for multipartite correlations in intrinsically genuine-multipartite
cryptographic scenarios. We prove that local measurements on
Greenberger-Horne-Zeilinger states, for all local dimension and number of
parts, can produce correlations that are fully genuine-multipartite nonlocal,
monogamous and with fully random outcomes. A key ingredient in our proof is a
multipartite chained Bell inequality detecting genuine-multipartite
nonlocality, which we introduce. Finally, we discuss the applications of our
results for intrinsically genuine-multipartite cryptographic protocols such as
device-independent secret sharing.
