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 Csiszár and Narayan. Building on the prior work of Tyagi for the twoterminal scenario, we derive a lower bound on the communication complexity, RSK, 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 RCO, is an upper bound on RSK. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain "Type S" condition, which is verifiable in polynomial time, guarantees that our lower bound on RSK meets the RCO upper bound. Thus, PIN models satisfying our condition are RSK-maximal, meaning that the upper bound RSK ≤ RCO holds with equality. This allows us to explicitly evaluate RSK for such PIN models. We also give several examples of PIN models that satisfy our Type S condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type S condition implies that communication from all terminals ("omnivocality") is needed for establishing a 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 S condition is satisfied. Counterexamples exist that show that the converse is not true in general for source models with four or more terminals.

For the multiterminal secret key agreement problem, new single-letter lower bounds are obtained on the public discussion rate required to achieve any given secret key rate below the secrecy capacity. The results apply to general source model without helpers or wiretapper's side information but can be strengthened for hypergraphical sources. In particular, for the pairwise independent network, the results give rise to a complete characterization of the maximum secret key rate achievable under a constraint on the total discussion rate.

In this paper, we address the problem of characterizing the instances of the multiterminal source model of Csiszár 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. We also give a relatively simple criterion for determining whether or not our condition holds for a given multiterminal source model.

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Abstract-In the multiterminal source model of Csiszár and Narayan, the communication complexity, R SK , for secret key (SK) generation is the minimum rate of communication required to achieve SK capacity. An obvious upper bound to R SK is given by R CO , which is the minimum rate of communication required for omniscience. In this paper we derive a better upper bound to R SK for the hypergraphical source model, which is a special instance of the multiterminal source model. The upper bound is based on the idea of fractional removal of hyperedges. It is further shown that this upper bound can be computed in polynomial time. We conjecture that our upper bound is tight. For the special case of a graphical source model, we also give an explicit lower bound on R SK . This bound, however, is not tight, as demonstrated by a counterexample.