  In the multiterminal source model of Csisz$\text{\'a}$r and Narayan, the
communication complexity, $R_{\text{SK}}$, for secret key (SK) generation is
the minimum rate of communication required to achieve SK capacity. An obvious
upper bound to $R_{\text{SK}}$ is given by $R_{\text{CO}}$, which is the
minimum rate of communication required for \emph{omniscience}. In this paper we
derive a better upper bound to $R_{\text{SK}}$ 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 $R_{\text{SK}}$. This bound,
however, is not tight, as demonstrated by a counterexample.
