  We presented assisted common information as a generalization of
G\'acs-K\"orner (GK) common information at ISIT 2010. The motivation for our
formulation was to improve upperbounds on the efficiency of protocols for
secure two-party sampling (which is a form of secure multi-party computation).
Our upperbound was based on a monotonicity property of a rate-region (called
the assisted residual information region) associated with the assisted common
information formulation. In this note we present further results. We explore
the connection of assisted common information with the Gray-Wyner system. We
show that the assisted residual information region and the Gray-Wyner region
are connected by a simple relationship: the assisted residual information
region is the increasing hull of the Gray-Wyner region under an affine map.
Several known relationships between GK common information and Gray-Wyner system
fall out as consequences of this. Quantities which arise in other source coding
contexts acquire new interpretations. In previous work we showed that assisted
common information can be used to derive upperbounds on the rate at which a
pair of parties can {\em securely sample} correlated random variables, given
correlated random variables from another distribution. Here we present an
example where the bound derived using assisted common information is much
better than previously known bounds, and in fact is tight. This example
considers correlated random variables defined in terms of standard variants of
oblivious transfer, and is interesting on its own as it answers a natural
question about these cryptographic primitives.
