Methods and apparatus are provided for more efficiently generating logic for implementing Exclusive OR (XOR) expressions. In one example, multiple centroids are generated and associated with main equations. Both the main equations and centroids are XOR expressions. In particular examples, main equations are associated with centroids that are closest in distance. Centroids can be modified to better match an associated subset of main equations. Centroids are generated and associated until the main equations are expressed as centroids and residue.