A Formalization of Polytime Functions

TL;DR

This paper formalizes Bellantoni and Cook's characterization of polynomial-time functions, providing a constructive proof, more precise bounds, and adapting it for cryptographic applications with a reusable library.

Contribution

It offers a fully constructive formalization of polytime functions on bitstrings, improving bounds and translation efficiency over prior work.

Findings

Formal proof of correctness and completeness

More precise bounding polynomials

Development of a reusable polytime function library

Abstract

We present a deep embedding of Bellantoni and Cook's syntactic characterization of polytime functions. We prove formally that it is correct and complete with respect to the original characterization by Cobham that required a bound to be proved manually. Compared to the paper proof by Bellantoni and Cook, we have been careful in making our proof fully contructive so that we obtain more precise bounding polynomials and more efficient translations between the two characterizations. Another difference is that we consider functions on bitstrings instead of functions on positive integers. This latter change is motivated by the application of our formalization in the context of formal security proofs in cryptography. Based on our core formalization, we have started developing a library of polytime functions that can be reused to build more complex ones.