Computable one-way functions on the reals

TL;DR

This paper constructs a total computable one-way function on the reals, addressing a major open problem by leveraging the halting problem's hardness, thus demonstrating such functions exist in the real domain.

Contribution

It provides the first explicit construction of a total computable one-way function on the reals, solving an open problem posed by Levin (2023).

Findings

Existence of a total computable one-way function on the reals.

Utilizes the hardness of the halting problem for construction.

Addresses a major open problem in computational complexity.

Abstract

A major open problem in computational complexity is the existence of a one-way function, namely a function from strings to strings which is computationally easy to compute but hard to invert. Levin (2023) formulated the notion of one-way functions from reals (infinite bit-sequences) to reals in terms of computability, and asked whether partial computable one-way functions exist. We give a strong positive answer using the hardness of the halting problem and exhibiting a total computable one-way function.