  This paper introduces the notion of exact common information, which is the
minimum description length of the common randomness needed for the exact
distributed generation of two correlated random variables $(X,Y)$. We introduce
the quantity $G(X;Y)=\min_{X\to W \to Y} H(W)$ as a natural bound on the exact
common information and study its properties and computation. We then introduce
the exact common information rate, which is the minimum description rate of the
common randomness for the exact generation of a 2-DMS $(X,Y)$. We give a
multiletter characterization for it as the limit $\bar{G}(X;Y)=\lim_{n\to
\infty}(1/n)G(X^n;Y^n)$. While in general $\bar{G}(X;Y)$ is greater than or
equal to the Wyner common information, we show that they are equal for the
Symmetric Binary Erasure Source. We do not know, however, if the exact common
information rate has a single letter characterization in general.
