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**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-t program generating a given string x is at most s.
In this work, we consider the more "robust" version of the time-bounded Kolmogorov complexity problem, referred to as the GapMINKT problem, where given a size bound s and a running time bound t, the goal is to determine whether there exists a poly(t,|x|)-time program of length s+O(log |x|) that generates x. We present the first non-trivial search-to-decision reduction R for the GapMINKT problem; R has a running-time bound of 2^{ε n} for any ε > 0 and additionally only queries its oracle on "thresholds" s of size s+O(log |x|). As such, we get that any algorithm with running-time (resp. circuit size) 2^{α s} poly(|x|,t,s) for solving GapMINKT (given an instance (x,t,s), yields an algorithm for finding a witness with running-time (resp. circuit size) 2^{(α+ε) s} poly(|x|,t,s).
Our second result is a polynomial-time search-to-decision reduction for the time-bounded Kolmogorov complexity problem in the average-case regime. Such a reduction was recently shown by Liu and Pass (FOCS'20), heavily relying on cryptographic techniques. Our reduction is more direct and additionally has the advantage of being length-preserving, and as such also applies in the exponential time/size regime.
A central component in both of these results is the use of Kolmogorov and Levin’s Symmetry of Information Theorem.

Noam Mazor and Rafael Pass. Search-To-Decision Reductions for Kolmogorov Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mazor_et_al:LIPIcs.CCC.2024.34, author = {Mazor, Noam and Pass, Rafael}, title = {{Search-To-Decision Reductions for Kolmogorov Complexity}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {34:1--34:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.34}, URN = {urn:nbn:de:0030-drops-204308}, doi = {10.4230/LIPIcs.CCC.2024.34}, annote = {Keywords: Kolmogorov complexity, search to decision} }

Document

**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

We demonstrate that under believable cryptographic hardness assumptions, Gap versions of standard meta-complexity problems, such as the Minimum Circuit Size Problem (MCSP) and the Minimum Time-Bounded Kolmogorov Complexity problem (MKTP) are not NP-complete w.r.t. Levin (i.e., witness-preserving many-to-one) reductions. In more detail:
- Assuming the existence of indistinguishability obfuscation, and subexponentially-secure one-way functions, an appropriate Gap version of MCSP is not NP-complete under randomized Levin-reductions.
- Assuming the existence of subexponentially-secure indistinguishability obfuscation, subexponentially-secure one-way functions and injective PRGs, an appropriate Gap version of MKTP is not NP-complete under randomized Levin-reductions.

Noam Mazor and Rafael Pass. Gap MCSP Is Not (Levin) NP-Complete in Obfustopia. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mazor_et_al:LIPIcs.CCC.2024.36, author = {Mazor, Noam and Pass, Rafael}, title = {{Gap MCSP Is Not (Levin) NP-Complete in Obfustopia}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {36:1--36:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.36}, URN = {urn:nbn:de:0030-drops-204322}, doi = {10.4230/LIPIcs.CCC.2024.36}, annote = {Keywords: Kolmogorov complexity, MCSP, Levin Reduction} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The Perebor (Russian for "brute-force search") conjectures, which date back to the 1950s and 1960s are some of the oldest conjectures in complexity theory. The conjectures are a stronger form of the NP ≠ P conjecture (which they predate) and state that for "meta-complexity" problems, such as the Time-bounded Kolmogorov complexity Problem, and the Minimum Circuit Size Problem, there are no better algorithms than brute force search.
In this paper, we disprove the non-uniform version of the Perebor conjecture for the Time-Bounded Kolmogorov complexity problem. We demonstrate that for every polynomial t(⋅), there exists of a circuit of size 2^{4n/5+o(n)} that solves the t(⋅)-bounded Kolmogorov complexity problem on every instance.
Our algorithm is black-box in the description of the Universal Turing Machine U employed in the definition of Kolmogorov Complexity and leverages the characterization of one-way functions through the hardness of the time-bounded Kolmogorov complexity problem of Liu and Pass (FOCS'20), and the time-space trade-off for one-way functions of Fiat and Naor (STOC'91). We additionally demonstrate that no such black-box algorithm can have circuit size smaller than 2^{n/2-o(n)}.
Along the way (and of independent interest), we extend the result of Fiat and Naor and demonstrate that any efficiently computable function can be inverted (with probability 1) by a circuit of size 2^{4n/5+o(n)}; as far as we know, this yields the first formal proof that a non-trivial circuit can invert any efficient function.

Noam Mazor and Rafael Pass. The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mazor_et_al:LIPIcs.ITCS.2024.80, author = {Mazor, Noam and Pass, Rafael}, title = {{The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {80:1--80:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.80}, URN = {urn:nbn:de:0030-drops-196088}, doi = {10.4230/LIPIcs.ITCS.2024.80}, annote = {Keywords: Kolmogorov complexity, perebor conjecture, function inversion} }

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**Published in:** LIPIcs, Volume 267, 4th Conference on Information-Theoretic Cryptography (ITC 2023)

Secret sharing schemes allow sharing a secret between a set of parties in a way that ensures that only authorized subsets of the parties learn the secret. Evolving secret sharing schemes (Komargodski, Naor, and Yogev [TCC '16]) allow achieving this end in a scenario where the parties arrive in an online fashion, and there is no a-priory bound on the number of parties.
An important complexity measure of a secret sharing scheme is the share size, which is the maximum number of bits that a party may receive as a share. While there has been a significant progress in recent years, the best constructions for both secret sharing and evolving secret sharing schemes have a share size that is exponential in the number of parties. On the other hand, the best lower bound, by Csirmaz [Eurocrypt '95], is sub-linear.
In this work, we give a tight lower bound on the share size of evolving secret sharing schemes. Specifically, we show that the sub-linear lower bound of Csirmaz implies an exponential lower bound on evolving secret sharing.

Noam Mazor. A Lower Bound on the Share Size in Evolving Secret Sharing. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 2:1-2:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mazor:LIPIcs.ITC.2023.2, author = {Mazor, Noam}, title = {{A Lower Bound on the Share Size in Evolving Secret Sharing}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {2:1--2:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-271-6}, ISSN = {1868-8969}, year = {2023}, volume = {267}, editor = {Chung, Kai-Min}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.2}, URN = {urn:nbn:de:0030-drops-183300}, doi = {10.4230/LIPIcs.ITC.2023.2}, annote = {Keywords: Secret sharing, Evolving secret sharing} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

A distribution is k-incompressible, Yao [FOCS '82], if no efficient compression scheme compresses it to less than k bits. While being a natural measure, its relation to other computational analogs of entropy such as pseudoentropy, Hastad, Impagliazzo, Levin, and Luby [SICOMP '99], and to other cryptographic hardness assumptions, was unclear.
We advance towards a better understating of this notion, showing that a k-incompressible distribution has (k-2) bits of next-block pseudoentropy, a refinement of pseudoentropy introduced by Haitner, Reingold, and Vadhan [SICOMP '13]. We deduce that a samplable distribution X that is (H(X)+2)-incompressible, implies the existence of one-way functions.

Iftach Haitner, Noam Mazor, and Jad Silbak. Incompressiblity and Next-Block Pseudoentropy. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 66:1-66:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{haitner_et_al:LIPIcs.ITCS.2023.66, author = {Haitner, Iftach and Mazor, Noam and Silbak, Jad}, title = {{Incompressiblity and Next-Block Pseudoentropy}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {66:1--66:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.66}, URN = {urn:nbn:de:0030-drops-175697}, doi = {10.4230/LIPIcs.ITCS.2023.66}, annote = {Keywords: incompressibility, next-block pseudoentropy, sparse languages} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Key-agreement protocols whose security is proven in the random oracle model are an important alternative to protocols based on public-key cryptography. In the random oracle model, the parties and the eavesdropper have access to a shared random function (an "oracle"), but the parties are limited in the number of queries they can make to the oracle. The random oracle serves as an abstraction for black-box access to a symmetric cryptographic primitive, such as a collision resistant hash. Unfortunately, as shown by Impagliazzo and Rudich [STOC '89] and Barak and Mahmoody [Crypto '09], such protocols can only guarantee limited secrecy: the key of any l-query protocol can be revealed by an O(l^2)-query adversary. This quadratic gap between the query complexity of the honest parties and the eavesdropper matches the gap obtained by the Merkle's Puzzles protocol of Merkle [CACM '78].
In this work we tackle a new aspect of key-agreement protocols in the random oracle model: their communication complexity. In Merkle's Puzzles, to obtain secrecy against an eavesdropper that makes roughly l^2 queries, the honest parties need to exchange Omega(l) bits. We show that for protocols with certain natural properties, ones that Merkle's Puzzle has, such high communication is unavoidable. Specifically, this is the case if the honest parties' queries are uniformly random, or alternatively if the protocol uses non-adaptive queries and has only two rounds. Our proof for the first setting uses a novel reduction from the set-disjointness problem in two-party communication complexity. For the second setting we prove the lower bound directly, using information-theoretic arguments.
Understanding the communication complexity of protocols whose security is proven (in the random-oracle model) is an important question in the study of practical protocols. Our results and proof techniques are a first step in this direction.

Iftach Haitner, Noam Mazor, Rotem Oshman, Omer Reingold, and Amir Yehudayoff. On the Communication Complexity of Key-Agreement Protocols. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haitner_et_al:LIPIcs.ITCS.2019.40, author = {Haitner, Iftach and Mazor, Noam and Oshman, Rotem and Reingold, Omer and Yehudayoff, Amir}, title = {{On the Communication Complexity of Key-Agreement Protocols}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {40:1--40:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.40}, URN = {urn:nbn:de:0030-drops-101335}, doi = {10.4230/LIPIcs.ITCS.2019.40}, annote = {Keywords: key agreement, random oracle, communication complexity, Merkle's puzzles} }

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