In all exact versions of scattering from a rough dielectric interface, six boundary unknowns must be found. In the Stratton-Chu equations, for example, the unknowns are the electric field (tangential and normal to the interface) and the boundary electric currents. Once determined, they are used to evaluate the three components of the scattered field and the three components of the transmitted field. The 6 x 6 operator matrix to be inverted consists of four 3 x 3 block matrices, two of which are diagonal. The other non-diagonal matrices describe the coupling. We describe a new formulation of this scattering problem in which the six boundary unknowns are the electric field and its normal derivative. In this formalism, three of the four 3 x 3 block matrices are diagonal. All the coupling resides in the fourth block.