Sketching is a software synthesis approach where the programmer develops a partial implementation — a sketch — and a separate specification of the desired functionality. The synthesizer then completes the sketch to behave like the specification. The correctness of the synthesized implementation is guaranteed by the compiler, which allows, among other benefits, rapid development of highly tuned implementations without the fear of introducing bugs. We develop SKETCH, a language for finite programs with linguistic support for sketching. Finite programs include many highperformance kernels, including cryptocodes. In contrast to prior synthesizers, which had to be equipped with domain-specific rules, SKETCH completes sketches by means of a combinatorial search based on generalized boolean satisfiability. Consequently, our combinatorial synthesizer is complete for the class of finite programs: it is guaranteed to complete any sketch in theory, and in practice has scaled to realis...