We show how to use dielectric thin films in order to reduce scattering and maximize broad-band absorption in arbitrary rough overcoated black surfaces. A rigourous calculation is developed thanks to an improved differential method, and the results are compared to experiment. Scattering is reduced by a factor 10 (down to 5.10(-3)), though the starting bare surface is a standard low-cost black paint (5.10(-2) scattering). Absorption can therefore be optimized up to 0.995.