The algebra of the Feistel-Toffoli construction

The process of replacing an arbitrary Boolean function by a bijective one, a fundamental tool in reversible computing and in cryptography, is interpreted algebraically as a particular instance of a certain group homomorphism from the [Formula: see text]-fold cartesian power of a group [Formula: see text] into the automorphism group of the free [Formula: see text]-set over the set [Formula: see text]. It is shown that this construction not only can be generalized from groups to monoids but, more generally, to internal categories in arbitrary finitely complete categories where it becomes a cartesian isomorphism between certain discrete fibrations.