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Documents authored by Myrisiotis, Dimitrios

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DOI: 10.4230/LIPIcs.FSTTCS.2021.7
One-Way Functions and a Conditional Variant of MKTP

Authors: Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.
Cite as

Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-Way Functions and a Conditional Variant of MKTP. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{allender_et_al:LIPIcs.FSTTCS.2021.7,
  author =	{Allender, Eric and Cheraghchi, Mahdi and Myrisiotis, Dimitrios and Tirumala, Harsha and Volkovich, Ilya},
  title =	{{One-Way Functions and a Conditional Variant of MKTP}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{7:1--7:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.7},
  URN =		{urn:nbn:de:0030-drops-155181},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.7},
  annote =	{Keywords: Kolmogorov complexity, KT Complexity, Minimum KT-complexity Problem, MKTP, Conditional KT Complexity, Minimum Conditional KT-complexity Problem, McKTP, one-way functions, OWFs, average-case hardness, pseudorandom generators, PRGs, pseudorandom functions, PRFs, distinguishers, learning algorithms, NP-completeness, reductions}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.23
One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

Authors: Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, and Yuichi Yoshida

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
For a size parameter s: ℕ → ℕ, the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}ⁿ → {0,1} (represented by a string of length N : = 2ⁿ) is at most a threshold s(n). A recent line of work exhibited "hardness magnification" phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant μ₁ > 0, if MCSP[2^{μ₁⋅ n}] cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^{1.01}, then P≠NP. In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: 1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute MCSP[2^{μ₂⋅n}] in time N^{1.99}, for some constant μ₂ > μ₁. 2) A non-deterministic (or parity) branching program of size o(N^{1.5}/log N) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nečiporuk method to MKTP, which previously appeared to be difficult. 3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least N^{1.5-o(1)}. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models. The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola 2019). En route, we obtain several related results: 1) There exists a (local) hitting set generator with seed length Õ(√N) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs. 2) Any read-once co-non-deterministic branching program computing MCSP must have size at least 2^Ω̃(N).
Cite as

Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, and Yuichi Yoshida. One-Tape Turing Machine and Branching Program Lower Bounds for MCSP. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cheraghchi_et_al:LIPIcs.STACS.2021.23,
  author =	{Cheraghchi, Mahdi and Hirahara, Shuichi and Myrisiotis, Dimitrios and Yoshida, Yuichi},
  title =	{{One-Tape Turing Machine and Branching Program Lower Bounds for MCSP}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{23:1--23:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.23},
  URN =		{urn:nbn:de:0030-drops-136681},
  doi =		{10.4230/LIPIcs.STACS.2021.23},
  annote =	{Keywords: Minimum Circuit Size Problem, Kolmogorov Complexity, One-Tape Turing Machines, Branching Programs, Lower Bounds, Pseudorandom Generators, Hitting Set Generators}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.15
Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

Authors: Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, and Igor C. Oliveira

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
The class 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢 consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class 𝒢. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]∘𝒢, for classes 𝒢 of functions with low communication complexity. Let R^(k)(𝒢) be the maximum k-party number-on-forehead randomized communication complexity of a function in 𝒢. Among other results, we show that: - The Generalized Inner Product function 𝖦𝖨𝖯^k_n cannot be computed in 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢 on more than 1/2+ε fraction of inputs for s = o(n²/{(k⋅4^k⋅R^(k)(𝒢)⋅log (n/ε)⋅log(1/ε))²}). This significantly extends the lower bounds against bipartite formulas obtained by [Avishay Tal, 2017]. As a corollary, we get an average-case lower bound for 𝖦𝖨𝖯^k_n against 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^{1.99}]∘𝖯𝖳𝖥^{k-1}, i.e., sub-quadratic-size de Morgan formulas with degree-(k-1) PTF (polynomial threshold function) gates at the bottom. - There is a PRG of seed length n/2 + O(√s⋅R^(2)(𝒢)⋅log(s/ε)⋅log(1/ε)) that ε-fools FORMULA[s]∘𝒢. For the special case of FORMULA[s]∘𝖫𝖳𝖥, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length O(n^{1/2}⋅s^{1/4}⋅log(n)⋅log(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where ε ≤ 1/n, complementing a recent result of [Ryan O'Donnell et al., 2019]. - There exists a randomized 2^{n-t}-time #SAT algorithm for 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢, where t = Ω(n/{√s⋅log²(s)⋅R^(2)(𝒢)})^{1/2}. In particular, this implies a nontrivial #SAT algorithm for 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖫𝖳𝖥. - The Minimum Circuit Size Problem is not in 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖷𝖮𝖱; thereby making progress on hardness magnification, in connection with results from [Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019]. On the algorithmic side, we show that the concept class 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖷𝖮𝖱 can be PAC-learned in time 2^O(n/log n).
Cite as

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, and Igor C. Oliveira. Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 15:1-15:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kabanets_et_al:LIPIcs.CCC.2020.15,
  author =	{Kabanets, Valentine and Koroth, Sajin and Lu, Zhenjian and Myrisiotis, Dimitrios and Oliveira, Igor C.},
  title =	{{Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{15:1--15:41},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.15},
  URN =		{urn:nbn:de:0030-drops-125673},
  doi =		{10.4230/LIPIcs.CCC.2020.15},
  annote =	{Keywords: de Morgan formulas, circuit lower bounds, satisfiability (SAT), pseudorandom generators (PRGs), learning, communication complexity, polynomial threshold functions (PTFs), parities}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.39
Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

Authors: Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most theta, for a given parameter theta. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N, requires - N^{3-o(1)}-size de Morgan formulas, improving the recent N^{2-o(1)} lower bound by Hirahara and Santhanam (CCC, 2017), - N^{2-o(1)}-size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and - 2^{Omega (N^{1/(d+2.01)})}-size depth-d AC^0 circuits, improving the superpolynomial lower bound by Allender et al. (SICOMP, 2006). The AC^0 lower bound stated above matches the best-known AC^0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an almost optimal lower bound of 2^{N^{1-o(1)}} for MCSP.
Cite as

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis. Circuit Lower Bounds for MCSP from Local Pseudorandom Generators. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cheraghchi_et_al:LIPIcs.ICALP.2019.39,
  author =	{Cheraghchi, Mahdi and Kabanets, Valentine and Lu, Zhenjian and Myrisiotis, Dimitrios},
  title =	{{Circuit Lower Bounds for MCSP from Local Pseudorandom Generators}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{39:1--39:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.39},
  URN =		{urn:nbn:de:0030-drops-106156},
  doi =		{10.4230/LIPIcs.ICALP.2019.39},
  annote =	{Keywords: minimum circuit size problem (MCSP), circuit lower bounds, pseudorandom generators (PRGs), local PRGs, de Morgan formulas, branching programs, constant depth circuits}
}
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