The algebra of Feistel-Toffoli schemes

TL;DR

This paper explores the algebraic structure of Feistel-Toffoli schemes, revealing a group-theoretic framework for reversible Boolean functions and their generalizations to monoids and categories, with implications for cryptography and reversible computing.

Contribution

It introduces a novel algebraic perspective on Boolean function bijections, extending the theory from groups to monoids and categories, and establishing a cartesian isomorphism in a broad categorical context.

Findings

Algebraic interpretation of bijective Boolean functions as group homomorphisms.

Generalization from groups to monoids and internal categories.

Establishment of a cartesian isomorphism in finitely complete categories.

Abstract

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 X-fold cartesian power of a group G into the automorphism group of the free G-set over the set X. 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.