Existence of classical one-way functions

Determine whether classical one-way functions exist in computational complexity; specifically, establish whether there is a total function f: {0,1}* -> {0,1}* that is efficiently computable (e.g., in polynomial time) but cannot be inverted with non-negligible success probability by any probabilistic efficient algorithm.

Background

The paper recalls the standard notion of one-way functions in classical computational complexity: finite maps that are easy to compute but hard to invert, even probabilistically. This foundational assumption underpins modern cryptographic primitives, yet its validity remains unresolved.

The authors contrast this longstanding complexity-theoretic open problem with their focus on the computability-theoretic setting of real functions, where they construct collision-resistant one-way functions under Levin’s framework. Their results do not address the classical complexity question, which they acknowledge remains open.

References

Modern cryptographic primitives rely on their existence, an unproven hypothesis which remains a long-standing open problem.

— Collision-resistant hash-shuffles on the reals  (2501.02604 - Barmpalias et al., 5 Jan 2025) in Section 1. Introduction

Hence, if $\eta$ non-negligible, there exists a PPT algorithm which can invert the hardcore predicate. We conclude the section by noting that it is not known whether one-way functions exist; in fact, showing their existence would imply $\text{P}\neq\text{NP}$. Furthermore, the existence of one-way functions alone also does not imply the existence of one-way permutations, and the latter is a stronger assumption.

— A Quantum/Classical Example Oracle Separation for Making Things Up  (2608.11648 - Chen, 12 Aug 2026) in Section 2.5, subsection “One-way functions”