On the circuit-size of inverses

TL;DR

This paper explores the circuit complexity of function inverses, demonstrating under certain complexity class separations that one-way functions exist with polynomial-size circuits, including total and surjective functions, some requiring non-uniformity.

Contribution

It strengthens previous results by showing the existence of total, length-preserving, one-way functions computable in polynomial time with polynomial-size circuits, assuming P^2^P ≠ Σ_2^P.

Findings

Existence of polynomial-time computable, length-preserving, one-way functions by circuit size.

Existence of polynomially balanced, surjective, one-way functions under complexity class separation.

Non-uniformity is necessary for some of these constructions.

Abstract

We reprove a result of Boppana and Lagarias: If Pi_2^P is different from Sigma_2^P then there exists a partial function f that is computable by a polynomial-size family of circuits, but no inverse of f is computable by a polynomial-size family of circuits. We strengthen this result by showing that there exist length-preserving total functions that are one-way by circuit size and that are computable in uniform polynomial time. We also prove, if Pi_2^P is different from Sigma_2^P, that there exist polynomially balanced total surjective functions that are one-way by circuit size; here non-uniformity is used.