  We consider a coded cooperative data exchange problem with the goal of
generating a secret key. Specifically, we investigate the number of public
transmissions required for a set of clients to agree on a secret key with
probability one, subject to the constraint that it remains private from an
eavesdropper.
  Although the problems are closely related, we prove that secret key
generation with fewest number of linear transmissions is NP-hard, while it is
known that the analogous problem in traditional cooperative data exchange can
be solved in polynomial time. In doing this, we completely characterize the
best possible performance of linear coding schemes, and also prove that linear
codes can be strictly suboptimal. Finally, we extend the single-key results to
characterize the minimum number of public transmissions required to generate a
desired integer number of statistically independent secret keys.
