The present paper gives a new formulation for rough surface scattering in terms of a stochastic integral equation which can be dealt with by means of stochastic functional approach. The random surface is assumed to be infinite and a homogeneous Gaussian random process. The random wave field is represented in the stochastic Floquet form due to the homogeneity of the surface, and in the non-Rayleigh form consisting of both upward and downward going scattered waves, as well as in the extended Voronovich form based on the consideration of the level-shift invariance. The stochastic integral equations of the first and the second kind are derived for the unknown surface source function which is a functional of the derivative or the increment of the surface profile function. It is also shown that the inhomogeneous term of the stochastic integral equation of the second kind automatically gives the solution of the Kirchhoff approximation for infinite surface.