  In this paper we generalize the notion of common information of two dependent
variables introduced by G\'acs & K\"orner. They defined common information as
the largest entropy rate of a common random variable two parties observing one
of the sources each can agree upon. It is well-known that their common
information captures only a limited form of dependence between the random
variables and is zero in most cases of interest. Our generalization, which we
call the Assisted Common Information system, takes into account almost-common
information ignored by G\'acs-K\"orner common information. In the assisted
common information system, a genie assists the parties in agreeing on a more
substantial common random variable; we characterize the trade-off between the
amount of communication from the genie and the quality of the common random
variable produced using a rate region we call the region of tension.
  We show that this region has an application in deriving upperbounds on the
efficiency of secure two-party sampling, which is a special case of secure
multi-party computation, a central problem in modern cryptography. Two parties
desire to produce samples of a pair of jointly distributed random variables
such that neither party learns more about the other's output than what its own
output reveals. They have access to a set up - correlated random variables
whose distribution is different from the desired distribution - and noiseless
communication. We present an upperbound on the rate at which a given set up can
be used to produce samples from a desired distribution by showing a
monotonicity property for the region of tension: a protocol between two parties
can only lower the tension between their views. Then, by calculating the bounds
on the region of tension of various pairs of correlated random variables, we
derive bounds on the rate of secure two-party sampling.
