The monoidal nature of the Feistel-Toffoli construction

AbstractThe Feistel-Toffoli construction of a bijective Boolean function out of an arbitrary one, a fundamental tool in reversible computing and in cryptography, has recently been analyzed (see [12]) to be a special instance of the construction of a monoid homomorphism from the X -fold cartesian power of a monoid M into the endomorphism monoid of the free M -set over the set X . It is the purpose of this note to show that this construction itself is in fact a genuine monoidal one. The generalization of the Feistel-Toffoli construction to internal categories in arbitrary finitely complete categories of [12] then becomes a special instance of this monoidal description.Mathematics Subject Classification (2020): 18M0518D4068Q09Key words: Convolution monoids and Hopf monoids in monoidal categoriesinternal categoryKleisli categoryspansFeistel schemeToffoli gate