It was shown earlier that some classes of three-dimensional contact problems can be mapped onto one-dimensional systems without loss of essential macroscopic information, thus allowing for immense acceleration of numerical simulations. The validity of this method of reduction of dimensionality has been strictly proven for contact of any axisymmetric bodies, both with and without adhesion. In [T. Geike and V. L. Popov, Phys. Rev. E 76, 036710 (2007)], it was shown that this method is valid "with empirical accuracy" for the simulation of contacts between randomly rough surfaces. In the present paper, we compare exact calculations of contact stiffness between elastic bodies with fractal rough surfaces (carried out by means of the boundary element method) with results of the corresponding one-dimensional model. Both calculations independently predict the contact stiffness as a function of the applied normal force to be a power law, with the exponent varying from 0.50 to 0.85, depending on the fractal dimension. The results strongly support the application of the method of reduction of dimensionality to a general class of randomly rough surfaces. The mapping onto a one-dimensional system drastically decreases the computation time.