The communication complexity of achieving SK capacity in a class of PIN models

The communication complexity of achieving secret key (SK) capacity in the multiterminal source model of Csisz$\'a$r and Narayan is the minimum rate of public communication required to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on the communication complexity, denoted by $R_{\text{SK}}$. A source model for which this upper bound is tight is called $R_{\text{SK}}$-maximal. In this paper, we establish a sufficient condition for $R_{\text{SK}}$-maximality within the class of pairwise independent network (PIN) models defined on hypergraphs. This allows us to compute $R_{\text{SK}}$ exactly within the class of PIN models satisfying this condition. On the other hand, we also provide a counterexample that shows that our condition does not in general guarantee $R_{\text{SK}}$-maximality for sources beyond PIN models.