HoTT-Crypt : A Study in Homotopy Type Theory based on Cryptography

This paper investigates a preliminary application of homotopy type theory in cryptography. It discusses specifying a cryptographic protocol using homotopy type theory which adds the notion of higher inductive type and univalence to Martin-Löf’s intensional type theory. A higher inductive type specification can act as a front-end mapped to a concrete cryptographic implementation in the universe. By having a higher inductive type front-end, we can encode domain-specific laws of the cryptographic implementation as higher-dimensional paths. The higher inductive type gives us a graphical computational model and can be used to extract functions from underlying concrete implementation. Using this model we can extend types to act as formal certificates guaranteeing on correctness properties of a cryptographic implementation.