On the Public Communication Needed to Achieve SK Capacity in the Multiterminal Source Model

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, R_SK, 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 R_CO, is an upper bound on R_SK. 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 R_SK meets the R_CO upper bound. Thus, the PIN models satisfying our condition are R_SK-maximal, indicating that the upper bound R_SK ≤ R_CO holds with equality. This allows us to explicitly evaluate R_SK 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 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 S condition is satisfied. However, for the source models with four or more terminals, counterexamples exist showing that the converse does not hold in general.