Method and apparatus for synthesizing high-performance linear finite state machines (LFSMs) such as linear feed-back shift registers (LFSRs) or cellular automata (CA). Given a characteristic polynomial for the circuit, the method obtains an original LFSR circuit (26) such as a type I or type II LFSR. Feedback connections within the original circuit are then determined (28). Subsequently, a number of transformations that shift the feedback connections can be applied in such a way that properties of the original circuit are preserved in a modified LFSR circuit (30). In particular, if the original circuit is represented by a primitive characteristic polynomial, the method preserved the maximum-length property of the original circuit in the modified circuit and enables the modified circuit to produce the same m-sequence as the original circuit. Through the various transformations, a modified LFSR circuit can be created that provides higher performance through shorter feedback connection lines, fewer levels of logic, and lower internal fan-out.