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One-Way Functions and Boundary Hardness of Randomized Time-Bounded Kolmogorov Complexity

Authors Yanyi Liu, Rafael Pass

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LIPIcs.ITCS.2026.97.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • One-way functions
  • Time-Bounded Kolmogorov Complexity
  • Worst-case to Average-case Reductions

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Abstract

We revisit the question of whether worst-case hardness of the time-bounded Kolmogorov complexity problem, MINK^{poly} - that is, determining whether a string is "structured" (i.e., K^t(x) < n-1) or "random" (i.e., K^{poly(t)} ≥ n-1) - suffices to imply the existence of one-way functions (OWF). Liu-Pass (CRYPTO'25) recently showed that worst-case hardness of a boundary version of MINK^{poly} - where, roughly speaking, the goal is to decide whether given an instance x, (a) x is K^poly-random (i.e., K^{poly(t)}(x) ≥ n-1), or just close to K^poly-random (i.e., K^{t}(x) < n-1 but K^{poly(t)} > n - log n) - characterizes OWF, but with either of the following caveats (1) considering a non-standard notion of probabilistic K^t, as opposed to the standard notion of K^t, or (2) assuming somewhat strong, and non-standard, derandomization assumptions.
In this paper, we present an alternative method for establishing their result which enables significantly weakening the caveats. First, we show that boundary hardness of the more standard randomized K^t problem suffices (where randomized K^t(x) is defined just like K^t(x) except that the program generating the string x may be randomized). 
As a consequence of this result, we can provide a characterization also in terms of just "plain" K^t under the most standard derandomization assumption (used to derandomize just BPP into P) - namely E ̸ ⊆ ioSIZE[2^{o(n)}].
Our proof relies on language compression schemes of Goldberg-Sipser (STOC'85); using the same technique, we also present the the first worst-case to average-case reduction for the exact MINK^{poly} problem (under the same standard derandomization assumption), improving upon Hirahara’s celebrated results (STOC'18, STOC'21) that only applied to a gap version of the MINK^{poly} problem, referred to as GapMINK^{poly}, where the goal is to decide whether K^t(x) ≤ n-O(log n)) or K^{poly(t)}(x) ≥ n-1 and under the same derandomization assumption.

Cite As Get BibTex

Yanyi Liu and Rafael Pass. One-Way Functions and Boundary Hardness of Randomized Time-Bounded Kolmogorov Complexity. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 97:1-97:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.97

Author Details

Yanyi Liu
  • Cornell Tech, New York, NY, USA
Rafael Pass
  • Cornell Tech, Technion, TAU, New York, NY, USA

Funding

  • Liu, Yanyi: Supported in part by NSF Award CNS 2149305.
  • Pass, Rafael: Supported in part by NSF Award CNS 2149305, AFOSR Award FA9550-24-1-0267, ISF Award 2338/23 and ERC Advanced Grant KolmoCrypt - 101142322. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government, the AFOSR, the European Union or the European Research Council Executive Agency.

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