TY - GEN
T1 - On One-Way Functions, the Worst-Case Hardness of Time-Bounded Kolmogorov Complexity, and Computational Depth
AU - Liu, Yanyi
AU - Pass, Rafael
N1 - Publisher Copyright:
© International Association for Cryptologic Research 2025.
PY - 2025
Y1 - 2025
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85211949360&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-78011-0_8
DO - 10.1007/978-3-031-78011-0_8
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AN - SCOPUS:85211949360
SN - 9783031780103
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 222
EP - 252
BT - Theory of Cryptography - 22nd International Conference, TCC 2024, Proceedings
A2 - Boyle, Elette
A2 - Boyle, Elette
A2 - Mahmoody, Mohammad
PB - Springer Science and Business Media Deutschland GmbH
T2 - 22nd Theory of Cryptography Conference, TCC 2024
Y2 - 2 December 2024 through 6 December 2024
ER -