  We consider a nonlinear evolution equation recently proposed to describe the
small-$x$ hadronic physics in the regime of very high gluon density. This is a
functional Fokker-Planck equation in terms of a classical random color source,
which represents the color charge density of the partons with large $x$. In the
saturation regime of interest, the coefficients of this equation must be known
to all orders in the source strength. In this first paper of a series of two,
we carefully derive the evolution equation, via a matching between classical
and quantum correlations, and set up the framework for the exact background
source calculation of its coefficients. We address and clarify many of the
subtleties and ambiguities which have plagued past attempts at an explicit
construction of this equation. We also introduce the physical interpretation of
the saturation regime at small $x$ as a Color Glass Condensate. In the second
paper we shall evaluate the expressions derived here, and compare them to known
results in various limits.
