  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 Csisz{\'a}r and Narayan. Building on the prior work of Tyagi for the
two-terminal scenario, we derive a lower bound on the communication complexity,
$R_{\text{SK}}$, 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 $R_{\text{CO}}$, is an upper bound on
$R_{\text{SK}}$. For the class of pairwise independent network (PIN) models
defined on uniform hypergraphs, we show that a certain "Type $\mathcal{S}$"
condition, which is verifiable in polynomial time, guarantees that our lower
bound on $R_{\text{SK}}$ meets the $R_{\text{CO}}$ upper bound. Thus, PIN
models satisfying our condition are $R_{\text{SK}}$-maximal, meaning that the
upper bound $R_{\text{SK}} \le R_{\text{CO}}$ holds with equality. This allows
us to explicitly evaluate $R_{\text{SK}}$ for such PIN models. We also give
several examples of PIN models that satisfy our Type $\mathcal S$ condition.
Finally, we prove that for an arbitrary multiterminal source model, a stricter
version of our Type $\mathcal S$ condition implies that communication from
\emph{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
$\mathcal S$ condition is satisfied. Counterexamples exist that show that the
converse is not true in general for source models with four or more terminals.
