Existing algorithms for finding restriction endonuclease recognition sites use brute-force algorithms which run in time 0(NM) where N is the number of nucleotides in the sequence under analysis and M is the total number of nucleotides in all the different sites being searched for. This paper presents a deterministic finite state machine algorithm which runs in time 0(N). Memory use can be as high as 0(M4) but a slight modification to the basic algorithm can impose a theoretical upper bound of 0(M) at the cost of some added complexity in the execution of the state machine. The algorithm can operate with a single pass through the sequence under analysis, with no need to back up or (for non-circular sequences) store more than a single input character at a time. This type of algorithm can be adapted to many pattern-matching tasks and is simple enough to implement in hardware that it could, for example, be built into a disk controller as part of a specialized database machine.