On One-Way Functions, the Worst-Case Hardness of Time-Bounded Kolmogorov Complexity, and Computational Depth

Yanyi Liu*, Rafael Pass

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Whether one-way functions (OWF) exist is arguably the most important problem in Cryptography, and beyond. While lots of candidate constructions of one-way functions are known, and recently also problems whose average-case hardness characterize the existence of OWFs have been demonstrated, the question of whether there exists some worst-case hard problem that characterizes the existence of one-way functions has remained open since their introduction in 1976. In this work, we present the first “OWF-complete” promise problem—a promise problem whose worst-case hardness w.r.t. BPP (resp. P/poly) is equivalent to the existence of OWFs secure against PPT (resp. nuPPT) algorithms. The problem is a variant of the Minimum Time-bounded Kolmogorov Complexity problem (MKtP[s] with a threshold s), where we condition on instances having small “computational depth”. We furthermore show that depending on the choice of the threshold s, this problem characterizes either “standard” (polynomially-hard) OWFs, or quasi polynomially- or subexponentially-hard OWFs. Additionally, when the threshold is sufficiently small (e.g., 2O(logn) or polylogn) then sublinear hardness of this problem suffices to characterize quasi-poly-nomial/sub-exponential OWFs. While our constructions are black-box, our analysis is non-black box; we additionally demonstrate that fully black-box constructions of OWF from the worst-case hardness of this problem are impossible. We finally show that, under Rudich’s conjecture, and standard derandomization assumptions, our problem is not inside coAM; as such, it yields the first candidate problem believed to be outside of AM∩coAM, or even SZK, whose worst case hardness implies the existence of OWFs.

Original languageEnglish
Title of host publicationTheory of Cryptography - 22nd International Conference, TCC 2024, Proceedings
EditorsElette Boyle, Elette Boyle, Mohammad Mahmoody
PublisherSpringer Science and Business Media Deutschland GmbH
Pages222-252
Number of pages31
ISBN (Print)9783031780103
DOIs
StatePublished - 2025
Externally publishedYes
Event22nd Theory of Cryptography Conference, TCC 2024 - Milan, Italy
Duration: 2 Dec 20246 Dec 2024

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume15364 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference22nd Theory of Cryptography Conference, TCC 2024
Country/TerritoryItaly
CityMilan
Period2/12/246/12/24

Funding

FundersFunder number
Algorand Centres of Excellence
Simons Institute for the Theory of Computing, University of California Berkeley
Algorand Foundation
National Science FoundationCNS 2149305, CNS-2128519, RI-1703846
Air Force Office of Scientific ResearchFA9550-23-1-0387, FA9550-18-1-0267, FA9550-23-1-0312
Defense Advanced Research Projects AgencyHR00110C0086

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