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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.35
URN: urn:nbn:de:0030-drops-143091
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14309/
Ren, Hanlin ;
Santhanam, Rahul
Hardness of KT Characterizes Parallel Cryptography
Abstract
A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity, K^t, is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures:
- We show, perhaps surprisingly, that the KT complexity is bounded-error average-case hard if and only if there exist one-way functions in constant parallel time (i.e. NC⁰). This result crucially relies on the idea of randomized encodings. Previously, a seminal work of Applebaum, Ishai, and Kushilevitz (FOCS'04; SICOMP'06) used the same idea to show that NC⁰-computable one-way functions exist if and only if logspace-computable one-way functions exist.
- Inspired by the above result, we present randomized average-case reductions among the NC¹-versions and logspace-versions of K^t complexity, and the KT complexity. Our reductions preserve both bounded-error average-case hardness and zero-error average-case hardness. To the best of our knowledge, this is the first reduction between the KT complexity and a variant of K^t complexity.
- We prove tight connections between the hardness of K^t complexity and the hardness of (the hardest) one-way functions. In analogy with the Exponential-Time Hypothesis and its variants, we define and motivate the Perebor Hypotheses for complexity measures such as K^t and KT. We show that a Strong Perebor Hypothesis for K^t implies the existence of (weak) one-way functions of near-optimal hardness 2^{n-o(n)}. To the best of our knowledge, this is the first construction of one-way functions of near-optimal hardness based on a natural complexity assumption about a search problem.
- We show that a Weak Perebor Hypothesis for MCSP implies the existence of one-way functions, and establish a partial converse. This is the first unconditional construction of one-way functions from the hardness of MCSP over a natural distribution.
- Finally, we study the average-case hardness of MKtP. We show that it characterizes cryptographic pseudorandomness in one natural regime of parameters, and complexity-theoretic pseudorandomness in another natural regime.
BibTeX - Entry
@InProceedings{ren_et_al:LIPIcs.CCC.2021.35,
author = {Ren, Hanlin and Santhanam, Rahul},
title = {{Hardness of KT Characterizes Parallel Cryptography}},
booktitle = {36th Computational Complexity Conference (CCC 2021)},
pages = {35:1--35:58},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-193-1},
ISSN = {1868-8969},
year = {2021},
volume = {200},
editor = {Kabanets, Valentine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14309},
URN = {urn:nbn:de:0030-drops-143091},
doi = {10.4230/LIPIcs.CCC.2021.35},
annote = {Keywords: one-way function, meta-complexity, KT complexity, parallel cryptography, randomized encodings}
}
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Keywords: |
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one-way function, meta-complexity, KT complexity, parallel cryptography, randomized encodings |
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Collection: |
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36th Computational Complexity Conference (CCC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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08.07.2021 |
