DNA strand displacement reactions (SDRs) provide a set of intelligent toolboxes for developing molecular computation. Whereas SDR-based logic gate circuits have achieved a high level of complexity, the scale-up for practical achievable computational tasks remains a hurdle. Switching circuits that were originally proposed by Shannon in 1938 and nowadays widely used in telecommunication represent an alternative and efficient means to realize fast-speed and high-bandwidth communication. Here we develop SDR-based DNA switching circuits (DSCs) for implementing digital computing. Using a routing strategy on a programmable DNA switch canvas, we show that arbitrary Boolean functions can be represented by DSCs and implemented with molecular switches with high computing speed. We further demonstrate the implementation of full-adder and square-rooting functions using DSCs, which only uses down to 1/4 DNA strands as compared with a dual-rail logic expression-based design. We expect that DSCs provide a design paradigm for digital computation with biomolecules.