The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csiszar and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, @math , defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by @math , is an upper bound on @math . For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain Type @math condition, which is verifiable in polynomial time, guarantees that our lower bound on @math meets the @math upper bound. Thus, the PIN models satisfying our condition are @math -maximal, indicating that the upper bound @math holds with equality. This allows us to explicitly evaluate @math for such PIN models. We also give several examples of PIN models that satisfy our Type @math condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type @math condition implies that communication from all terminals (omnivocality) is needed for establishing an SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type @math condition is satisfied. However, for the source models with four or more terminals, counterexamples exist showing that the converse does not hold in general.
Turning our attention to the topic of omnivocality originally studied in our paper @cite_16 , @cite_25 have recently obtained some new results. In particular, their Theorem 5 gives a sufficient condition for when a particular terminal communicate in any SK-capacity-achieving protocol. Our original sufficient condition for omnivocality [Theorem 4] MKS14 (Theorem in this paper) can now be obtained as a consequence of 's Theorem 5. In addition, Theorem 4 of @cite_25 provides a sufficient condition that guarantees the existence of an SK-capacity-achieving protocol within which a given terminal can remain silent.
@cite_16 @cite_25
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In this paper, we address the problem of characterizing the instances of the multiterminal source model of Csiszar and Narayan in which communication from all terminals is needed for establishing a secret key of maximum rate. We give an information-theoretic sufficient condition for identifying such instances. We believe that our sufficient condition is in fact an exact characterization, but we are only able to prove this in the case of the three-terminal source model. The problem of when all terminals must talk to achieve the secrecy capacity in the multiterminal source model is investigated. Two conditions under which respectively a given terminal does not need to and must talk to achieve the secrecy capacity are characterized. The cases when all terminals must talk to achieve secrecy capacity are shown to be many more than those conjectured in [1] for systems with four or more terminals. There is a gap between the above two conditions, in which whether a given terminal need to talk is not clear. A conjecture is further made in order to narrow down the gap.