18 Search Results for "Oliveira, Igor Carboni"

Document
DOI: 10.4230/LIPIcs.STACS.2026.23
Simple Circuit Extensions for XOR in PTIME

Authors: Marco Carmosino, Ngu Dang, and Tim Jackman

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
The Minimum Circuit Size Problem for Partial Functions (MCSP^*) is hard assuming the Exponential Time Hypothesis (ETH) (Ilango, 2020). This breakthrough result leveraged a characterization of the optimal {∧, ∨, ¬} circuits for n-bit OR (OR_n) and a reduction from the partial f-Simple Extension Problem where f = OR_n. It remains open to extend that reduction to show ETH-hardness of total MCSP. However, Ilango observed that the total f-Simple Extension Problem is easy whenever f is computed by read-once formulas (like OR_n). Therefore, extending Ilango’s proof to total MCSP would require replacing OR_n with a more complex but similarly well-understood Boolean function. This work shows that the f-Simple Extension problem remains easy when f is the next natural candidate: XOR_n. We first develop a fixed-parameter tractable algorithm for the f-Simple Extension Problem that is efficient whenever the optimal circuits for f are (1) linear in size, (2) polynomially "few" and efficiently enumerable in the truth-table size (up to isomorphism and permutation of inputs), and (3) all have constant bounded fan-out. XOR_n satisfies all three of these conditions. When ¬ gates count towards circuit size, optimal XOR_n circuits are binary trees of n-1 subcircuits computing (¬)XOR₂ (Kombarov, 2011). We extend this characterization when ¬ gates do not contribute the circuit size. Thus, the XOR-Simple Extension Problem is in polynomial time under both measures of circuit complexity. We conclude by discussing conjectures about the complexity of the f-Simple Extension problem for each explicit function f with known and unrestricted circuit lower bounds over the DeMorgan basis. Examining the conditions under which our Simple Extension Solver is efficient, we argue that multiplexer functions (MUX) are the most promising candidate for ETH-hardness of a Simple Extension Problem, towards proving ETH-hardness of total MCSP.
Cite as

Marco Carmosino, Ngu Dang, and Tim Jackman. Simple Circuit Extensions for XOR in PTIME. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{carmosino_et_al:LIPIcs.STACS.2026.23,
  author =	{Carmosino, Marco and Dang, Ngu and Jackman, Tim},
  title =	{{Simple Circuit Extensions for XOR in PTIME}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.23},
  URN =		{urn:nbn:de:0030-drops-255127},
  doi =		{10.4230/LIPIcs.STACS.2026.23},
  annote =	{Keywords: Minimum Circuit Size Problem, Circuit Lower Bounds, Exponential Time Hypothesis}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.97
One-Way Functions and Boundary Hardness of Randomized Time-Bounded Kolmogorov Complexity

Authors: Yanyi Liu and Rafael Pass

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

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

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)

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@InProceedings{liu_et_al:LIPIcs.ITCS.2026.97,
  author =	{Liu, Yanyi and Pass, Rafael},
  title =	{{One-Way Functions and Boundary Hardness of Randomized Time-Bounded Kolmogorov Complexity}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{97:1--97:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.97},
  URN =		{urn:nbn:de:0030-drops-253849},
  doi =		{10.4230/LIPIcs.ITCS.2026.97},
  annote =	{Keywords: One-way functions, Time-Bounded Kolmogorov Complexity, Worst-case to Average-case Reductions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.112
Lower Bounds Beyond DNF of Parities

Authors: Artur Riazanov, Anastasia Sofronova, and Dmitry Sokolov

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We consider a subclass of AC⁰[2] circuits that simultaneously captures DNF∘Xor and depth-3 AC⁰ circuits. For this class we show a technique for proving lower bounds inspired by the top-down approach. We give lower bounds for the middle slice function, inner product function, and affine dispersers.
Cite as

Artur Riazanov, Anastasia Sofronova, and Dmitry Sokolov. Lower Bounds Beyond DNF of Parities. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 112:1-112:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{riazanov_et_al:LIPIcs.ITCS.2026.112,
  author =	{Riazanov, Artur and Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{Lower Bounds Beyond DNF of Parities}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{112:1--112:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.112},
  URN =		{urn:nbn:de:0030-drops-253996},
  doi =		{10.4230/LIPIcs.ITCS.2026.112},
  annote =	{Keywords: boolean circuits, top-down, unpredictability}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.31
Towards Free Lunch Derandomization from Necessary Assumptions (And OWFs)

Authors: Marshall Ball, Lijie Chen, and Roei Tell

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
The question of optimal derandomization, introduced by Doron et. al (JACM 2022), garnered significant recent attention. Works in recent years showed conditional superfast derandomization algorithms, as well as conditional impossibility results, and barriers for obtaining superfast derandomization using certain black-box techniques. Of particular interest is the extreme high-end, which focuses on "free lunch" derandomization, as suggested by Chen and Tell (FOCS 2021). This is derandomization that incurs essentially no time overhead, and errs only on inputs that are infeasible to find. Constructing such algorithms is challenging, and so far there have not been any results following the one in their initial work. In their result, their algorithm is essentially the classical Nisan-Wigderson generator, and they relied on an ad-hoc assumption asserting the existence of a function that is non-batch-computable over all polynomial-time samplable distributions. In this work we deduce free lunch derandomization from a variety of natural hardness assumptions. In particular, we do not resort to non-batch-computability, and the common denominator for all of our assumptions is hardness over all polynomial-time samplable distributions, which is necessary for the conclusion. The main technical components in our proofs are constructions of new and superfast targeted generators, which completely eliminate the time overheads that are inherent to all previously known constructions. In particular, we present an alternative construction for the targeted generator by Chen and Tell (FOCS 2021), which is faster than the original construction, and also more natural and technically intuitive. These contributions significantly strengthen the evidence for the possibility of free lunch derandomization, distill the required assumptions for such a result, and provide the first set of dedicated technical tools that are useful for studying the question.
Cite as

Marshall Ball, Lijie Chen, and Roei Tell. Towards Free Lunch Derandomization from Necessary Assumptions (And OWFs). In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ball_et_al:LIPIcs.CCC.2025.31,
  author =	{Ball, Marshall and Chen, Lijie and Tell, Roei},
  title =	{{Towards Free Lunch Derandomization from Necessary Assumptions (And OWFs)}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{31:1--31:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.31},
  URN =		{urn:nbn:de:0030-drops-237259},
  doi =		{10.4230/LIPIcs.CCC.2025.31},
  annote =	{Keywords: Pseudorandomness, Derandomization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.36
Lifting with Colourful Sunflowers

Authors: Susanna F. de Rezende and Marc Vinyals

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We show that a generalization of the DAG-like query-to-communication lifting theorem, when proven using sunflowers over non-binary alphabets, yields lower bounds on the monotone circuit complexity and proof complexity of natural functions and formulas that are better than previously known results obtained using the approximation method. These include an n^Ω(k) lower bound for the clique function up to k ≤ n^{1/2-ε}, and an exp(Ω(n^{1/3-ε})) lower bound for a function in P.
Cite as

Susanna F. de Rezende and Marc Vinyals. Lifting with Colourful Sunflowers. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2025.36,
  author =	{de Rezende, Susanna F. and Vinyals, Marc},
  title =	{{Lifting with Colourful Sunflowers}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{36:1--36:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.36},
  URN =		{urn:nbn:de:0030-drops-237303},
  doi =		{10.4230/LIPIcs.CCC.2025.36},
  annote =	{Keywords: lifting, sunflower, clique, colouring, monotone circuit, cutting planes}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.46
On the Structure of Learnability Beyond P/Poly

Authors: Ninad Rajgopal and Rahul Santhanam

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
Motivated by the goal of showing stronger structural results about the complexity of learning, we study the learnability of strong concept classes beyond P/poly, such as PSPACE/poly and EXP/poly. We show the following: 1) (Unconditional Lower Bounds for Learning) Building on [Adam R. Klivans et al., 2013], we prove unconditionally that BPE/poly cannot be weakly learned in polynomial time over the uniform distribution, even with membership and equivalence queries. 2) (Robustness of Learning) For the concept classes EXP/poly and PSPACE/poly, we show unconditionally that worst-case and average-case learning are equivalent, that PAC-learnability and learnability over the uniform distribution are equivalent, and that membership queries do not help in either case. 3) (Reducing Succinct Search to Decision for Learning) For the decision problems R_{Kt} and R_{KS} capturing the complexity of learning EXP/poly and PSPACE/poly respectively, we show a succinct search to decision reduction: for each of these problems, the problem is in BPP iff there is a probabilistic polynomial-time algorithm computing circuits encoding proofs for positive instances of the problem. This is shown via a more general result giving succinct search to decision results for PSPACE, EXP and NEXP, which might be of independent interest. 4) (Implausibility of Oblivious Strongly Black-Box Reductions showing NP-hardness of learning NP/poly) We define a natural notion of hardness of learning with respect to oblivious strongly black-box reductions. We show that learning PSPACE/poly is PSPACE-hard with respect to oblivious strongly black-box reductions. On the other hand, if learning NP/poly is NP-hard with respect to oblivious strongly black-box reductions, the Polynomial Hierarchy collapses.
Cite as

Ninad Rajgopal and Rahul Santhanam. On the Structure of Learnability Beyond P/Poly. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 46:1-46:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{rajgopal_et_al:LIPIcs.APPROX/RANDOM.2021.46,
  author =	{Rajgopal, Ninad and Santhanam, Rahul},
  title =	{{On the Structure of Learnability Beyond P/Poly}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{46:1--46:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.46},
  URN =		{urn:nbn:de:0030-drops-147395},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.46},
  annote =	{Keywords: Hardness of Learning, Oracle Circuit Classes, Succinct Search, Black-Box Reductions}
}
Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2021.89
Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract)

Authors: Yuval Filmus, Or Meir, and Avishay Tal

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of O(p²) under a random restriction that leaves each variable alive independently with probability p [SICOMP, 1998]. Using this result, he gave an Ω̃(n³) formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this work, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization - random projections. Based on our shrinkage results, we obtain an Ω̃(n³) formula size lower bound for an explicit function computed in AC⁰. This improves upon the best known formula size lower bounds for AC⁰, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function’s structure so that the function maintains structure even under projection - using such projections is necessary, as standard random restrictions simplify AC⁰ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on the proof of Håstad for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of p-random restrictions, our proof can be used as an exposition of Håstad’s result.
Cite as

Yuval Filmus, Or Meir, and Avishay Tal. Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 89:1-89:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{filmus_et_al:LIPIcs.ITCS.2021.89,
  author =	{Filmus, Yuval and Meir, Or and Tal, Avishay},
  title =	{{Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{89:1--89:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.89},
  URN =		{urn:nbn:de:0030-drops-136281},
  doi =		{10.4230/LIPIcs.ITCS.2021.89},
  annote =	{Keywords: De Morgan formulas, KRW Conjecture, shrinkage, random restrictions, random projections, bounded depth circuits, constant depth circuits, formula complexity}
}
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
DOI: 10.4230/LIPIcs.ITCS.2020.19
Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Authors: Fedor Part and Iddo Tzameret

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted Res(lin_R), this refutation system operates with disjunctions of linear equations with boolean variables over a ring R, to refute unsatisfiable sets of such disjunctions. Beginning in the work of [Ran Raz and Iddo Tzameret, 2008], through the work of [Dmitry Itsykson and Dmitry Sokolov, 2014] which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. [Jan Krajícek, 2017; Dmitry Itsykson and Dmitry Sokolov, 2014; Jan Krajícek and Igor Carboni Oliveira, 2018; Michal Garlik and Lezsek Kołodziejczyk, 2018]) made it evident that establishing lower bounds against general Res(lin_R) refutations is a challenging and interesting task since the system captures a "minimal" extension of resolution with counting gates for which no super-polynomial lower bounds are known to date. We provide the first super-polynomial size lower bounds on general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular we prove that the subset-sum principle 1+ x_1 + ̇s +2^n x_n = 0 requires refutations of exponential-size over ℚ. Our proof technique is nontrivial and novel: roughly speaking, we show that under certain conditions every refutation of a subset-sum instance f=0, where f is a linear polynomial over ℚ, must pass through a fat clause containing an equation f=α for each α in the image of f under boolean assignments. We develop a somewhat different approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on n variables, hence also separating tree-like from dag-like refutations over the rationals. We then turn to the finite fields regime, showing that the work of Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014] who obtained tree-like lower bounds over ?_2 can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (i) for every pair of distinct primes p,q, there exist CNF formulas with short tree-like refutations in Res(lin_{?_p}) that require exponential-size tree-like Res(lin_{?_q}) refutations; (ii) random k-CNF formulas require exponential-size tree-like Res(lin_{?_p}) refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like Res(lin_?) refutations of the pigeonhole principle, for every field ?.
Cite as

Fedor Part and Iddo Tzameret. Resolution with Counting: Dag-Like Lower Bounds and Different Moduli. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 19:1-19:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{part_et_al:LIPIcs.ITCS.2020.19,
  author =	{Part, Fedor and Tzameret, Iddo},
  title =	{{Resolution with Counting: Dag-Like Lower Bounds and Different Moduli}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{19:1--19:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.19},
  URN =		{urn:nbn:de:0030-drops-117041},
  doi =		{10.4230/LIPIcs.ITCS.2020.19},
  annote =	{Keywords: Proof complexity, concrete lower bounds, resolution, satisfiability, combinatorics}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.68
Pseudorandomness and the Minimum Circuit Size Problem

Authors: Rahul Santhanam

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
We explore the possibility of basing one-way functions on the average-case hardness of the fundamental Minimum Circuit Size Problem (MCSP[s]), which asks whether a Boolean function on n bits specified by its truth table has circuits of size s(n). 1) (Pseudorandomness from Zero-Error Average-Case Hardness) We show that for a given size function s, the following are equivalent: Pseudorandom distributions supported on strings describable by s(O(n))-size circuits exist; Hitting sets supported on strings describable by s(O(n))-size circuits exist; MCSP[s(O(n))] is zero-error average-case hard. Using similar techniques, we show that Feige’s hypothesis for random k-CNFs implies that there is a pseudorandom distribution (with constant error) supported entirely on satisfiable formulas. Underlying our results is a general notion of semantic sampling, which might be of independent interest. 2) (A New Conjecture) In analogy to a known universal construction of succinct hitting sets against arbitrary polynomial-size adversaries, we propose the Universality Conjecture: there is a universal construction of succinct pseudorandom distributions against arbitrary polynomial-size adversaries. We show that under the Universality Conjecture, the following are equivalent: One-way functions exist; Natural proofs useful against sub-exponential size circuits do not exist; Learning polynomial-size circuits with membership queries over the uniform distribution is hard; MCSP[2^(ε n)] is zero-error hard on average for some ε > 0; Cryptographic succinct hitting set generators exist. 3) (Non-Black-Box Results) We show that for weak circuit classes ℭ against which there are natural proofs [Alexander A. Razborov and Steven Rudich, 1997], pseudorandom functions secure against poly-size circuits in ℭ imply superpolynomial lower bounds in P against poly-size circuits in ℭ. We also show that for a certain natural variant of MCSP, there is a polynomial-time reduction from approximating the problem well in the worst case to solving it on average. These results are shown using non-black-box techniques, and in the first case we show that there is no black-box proof of the result under standard crypto assumptions.
Cite as

Rahul Santhanam. Pseudorandomness and the Minimum Circuit Size Problem. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 68:1-68:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{santhanam:LIPIcs.ITCS.2020.68,
  author =	{Santhanam, Rahul},
  title =	{{Pseudorandomness and the Minimum Circuit Size Problem}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{68:1--68:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.68},
  URN =		{urn:nbn:de:0030-drops-117532},
  doi =		{10.4230/LIPIcs.ITCS.2020.68},
  annote =	{Keywords: Minimum Circuit Size Problem, Pseudorandomness, Average-case Complexity, Natural Proofs, Universality Conjecture}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.70
Beyond Natural Proofs: Hardness Magnification and Locality

Authors: Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Ján Pich, Ninad Rajgopal, and Rahul Santhanam

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
Hardness magnification reduces major complexity separations (such as EXP ⊈ NC^1) to proving lower bounds for some natural problem Q against weak circuit models. Several recent works [Igor Carboni Oliveira and Rahul Santhanam, 2018; Dylan M. McKay et al., 2019; Lijie Chen and Roei Tell, 2019; Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019; Igor Carboni Oliveira, 2019; Lijie Chen et al., 2019] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than Q, while Q itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: - Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [Alexander A. Razborov and Steven Rudich, 1997]? - Can we adapt known lower bound techniques to establish the desired lower bound for Q? We establish that some instantiations of hardness magnification overcome the natural proofs barrier in the following sense: slightly superlinear-size circuit lower bounds for certain versions of the minimum circuit size problem MCSP imply the non-existence of natural proofs. As a corollary of our result, we show that certain magnification theorems not only imply strong worst-case circuit lower bounds but also rule out the existence of efficient learning algorithms. Hardness magnification might sidestep natural proofs, but we identify a source of difficulty when trying to adapt existing lower bound techniques to prove strong lower bounds via magnification. This is captured by a locality barrier: existing magnification theorems unconditionally show that the problems Q considered above admit highly efficient circuits extended with small fan-in oracle gates, while lower bound techniques against weak circuit models quite often easily extend to circuits containing such oracles. This explains why direct adaptations of certain lower bounds are unlikely to yield strong complexity separations via hardness magnification.
Cite as

Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Ján Pich, Ninad Rajgopal, and Rahul Santhanam. Beyond Natural Proofs: Hardness Magnification and Locality. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 70:1-70:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2020.70,
  author =	{Chen, Lijie and Hirahara, Shuichi and Oliveira, Igor C. and Pich, J\'{a}n and Rajgopal, Ninad and Santhanam, Rahul},
  title =	{{Beyond Natural Proofs: Hardness Magnification and Locality}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{70:1--70:48},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.70},
  URN =		{urn:nbn:de:0030-drops-117550},
  doi =		{10.4230/LIPIcs.ITCS.2020.70},
  annote =	{Keywords: Hardness Magnification, Natural Proofs, Minimum Circuit Size Problem, Circuit Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.23
Parity Helps to Compute Majority

Authors: Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC^0[oplus] circuits. Razborov [Alexander A. Razborov, 1987] and Smolensky [Roman Smolensky, 1987; Roman Smolensky, 1993] showed that Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/2(d-1)})}. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-d AC^0[oplus] circuits of size 2^{O~(n^{1/(d-1)})}. This gap between upper and lower bounds has stood for nearly three decades. Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large d. We show for d >= 5 that any symmetric function can be computed with depth-d AC^0[oplus] circuits of size exp(O~(n^{2/3 * 1/(d-4)})). Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for d >= 3 Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/(2d-4)})}. For depths d <= 4, we are able to refine our techniques to get almost-optimal bounds: the depth-3 AC^0[oplus] circuit size of Majority is 2^{Theta~(n^{1/2})}, while its depth-4 AC^0[oplus] circuit size is 2^{Theta~(n^{1/4})}.
Cite as

Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan. Parity Helps to Compute Majority. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.23,
  author =	{Oliveira, Igor Carboni and Santhanam, Rahul and Srinivasan, Srikanth},
  title =	{{Parity Helps to Compute Majority}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{23:1--23:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.23},
  URN =		{urn:nbn:de:0030-drops-108453},
  doi =		{10.4230/LIPIcs.CCC.2019.23},
  annote =	{Keywords: Computational Complexity, Boolean Circuits, Lower Bounds, Parity, Majority}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.27
Hardness Magnification near State-Of-The-Art Lower Bounds

Authors: Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity <= s_1(N) from instances of complexity >= s_2(N), and N = 2^n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin’s notion of time-bounded Kolmogorov complexity. (In our results, the parameters s_1(N) and s_2(N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds. Theorem. There exists a universal constant c >= 1 for which the following hold. If there exists epsilon > 0 such that for every small enough beta > 0 (1) Gap-MCSP[2^{beta n}/c n, 2^{beta n}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U_2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U_2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP !subseteq NC^1 would follow via hardness magnification.
Cite as

Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness Magnification near State-Of-The-Art Lower Bounds. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 27:1-27:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.27,
  author =	{Oliveira, Igor Carboni and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{Hardness Magnification near State-Of-The-Art Lower Bounds}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{27:1--27:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.27},
  URN =		{urn:nbn:de:0030-drops-108494},
  doi =		{10.4230/LIPIcs.CCC.2019.27},
  annote =	{Keywords: Circuit Complexity, Minimum Circuit Size Problem, Kolmogorov Complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.32
Randomness and Intractability in Kolmogorov Complexity

Authors: Igor Carboni Oliveira

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

Abstract
We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin’s notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability. This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time. Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.
Cite as

Igor Carboni Oliveira. Randomness and Intractability in Kolmogorov Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira:LIPIcs.ICALP.2019.32,
  author =	{Oliveira, Igor Carboni},
  title =	{{Randomness and Intractability in Kolmogorov Complexity}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{32:1--32:14},
  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.2019.32},
  URN =		{urn:nbn:de:0030-drops-106087},
  doi =		{10.4230/LIPIcs.ICALP.2019.32},
  annote =	{Keywords: computational complexity, randomness, circuit lower bounds, Kolmogorov complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.18
Expander-Based Cryptography Meets Natural Proofs

Authors: Igor Carboni Oliveira, Rahul Santhanam, and Roei Tell

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

Abstract
We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are: 1) We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbour function has low circuit complexity might compromise the security of Goldreich's PRG and OWF in certain settings. 2) We show that the security of Goldreich's PRG and OWF is closely related to two other long-standing problems: Specifically, to the existence of unbalanced lossless expanders with low-complexity neighbor function, and to limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits. We prove two types of technical results that support the above conceptual messages. First, we unconditionally break Goldreich's PRG when instantiated with a specific expander (whose existence we prove), for a class of predicates that match the parameters of the currently-best "hard" candidates, in the regime of quasi-polynomial stretch. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's generator in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's generator is insecure in the regime of a large polynomial stretch, regardless of the predicate used.
Cite as

Igor Carboni Oliveira, Rahul Santhanam, and Roei Tell. Expander-Based Cryptography Meets Natural Proofs. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carbonioliveira_et_al:LIPIcs.ITCS.2019.18,
  author =	{Carboni Oliveira, Igor and Santhanam, Rahul and Tell, Roei},
  title =	{{Expander-Based Cryptography Meets Natural Proofs}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{18:1--18:14},
  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.2019.18},
  URN =		{urn:nbn:de:0030-drops-101112},
  doi =		{10.4230/LIPIcs.ITCS.2019.18},
  annote =	{Keywords: Pseudorandom Generators, One-Way Functions, Expanders, Circuit Complexity}
}
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