  Hypergraphical sources are a natural class of sources for secret key
generation, within which different subsets of terminals sharing secrets are
allowed to discuss publicly in order to agree upon a global secret key. While
their secrecy capacity, i.e., the maximum rate of a secret key that can be
agreed upon by the entire set of terminals, is well-understood, what remains
open is the maximum rate of a secret key that can be generated when there is a
restriction on the overall rate of public discussion allowed. In this work, we
obtain a family of explicitly computable upper bounds on the number of bits of
secret key that can be generated per bit of public discussion. These upper
bounds are derived using a lamination technique based on the submodularity of
the entropy function. In particular, a specific instance of these upper bounds,
called the edge-partition bound, is shown to be tight for the pairwise
independent network model, a special case of the hypergraphical source. The
secret key generation scheme achieving this upper bound is the tree-packing
protocol of Nitinawarat et al., thereby resolving in the affirmative the
discussion rate optimality of the tree packing protocol.
