  We explore the duality between the simulation and extraction of secret
correlations in light of a similar well-known operational duality between the
two notions of common information due to Wyner, and G\'acs and K\"orner. For
the inverse problem of simulating a tripartite noisy correlation from noiseless
secret key and unlimited public communication, we show that Winter's (2005)
result for the key cost in terms of a conditional version of Wyner's common
information can be simply reexpressed in terms of the existence of a bipartite
protocol monotone. For the forward problem of key distillation from noisy
correlations, we construct simple distributions for which the conditional
G\'acs and K\"orner common information achieves a tight bound on the secret key
rate. We conjecture that this holds in general for non-communicative key
agreement models. We also comment on the interconvertibility of secret
correlations under local operations and public communication.
