One well known method of generating key stream sequences for stream ciphers is to combine the outputs of several Linear Feedback Shift Registers (LFSR) using a combining Boolean function. Here we concentrate on the design of good combining Boolean functions. We provide resilient Boolean functions with currently best known nonlinearity. These functions were not known earlier and the problems of their existence were posed as open questions in literature. Some of the functions we construct here achieve the provable upper bound on nonlinearity for resilient Boolean functions. Our technique interlinks mathematical results with classical computer search.