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DOI: 10.4230/LIPIcs.CCC.2024.10

Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

Authors: Xin Li and Yan Zhong

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

Abstract

Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation. This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include: - An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works. - An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).

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Xin Li and Yan Zhong. Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.CCC.2024.10,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{10:1--10:14},
  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.2024.10},
  URN =		{urn:nbn:de:0030-drops-204060},
  doi =		{10.4230/LIPIcs.CCC.2024.10},
  annote =	{Keywords: Randomness Extractors, Affine, Read-once Linear Branching Programs, Low-degree polynomials, AC⁰ circuits}
}

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DOI: 10.4230/LIPIcs.CCC.2024.17

Local Enumeration and Majority Lower Bounds

Authors: Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard

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

Abstract

Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics.

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Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard. Local Enumeration and Majority Lower Bounds. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gurumukhani_et_al:LIPIcs.CCC.2024.17,
  author =	{Gurumukhani, Mohit and Paturi, Ramamohan and Pudl\'{a}k, Pavel and Saks, Michael and Talebanfard, Navid},
  title =	{{Local Enumeration and Majority Lower Bounds}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{17:1--17:25},
  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.2024.17},
  URN =		{urn:nbn:de:0030-drops-204136},
  doi =		{10.4230/LIPIcs.CCC.2024.17},
  annote =	{Keywords: Depth 3 circuits, k-CNF satisfiability, Circuit lower bounds, Majority function}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.23

Exponential Separation Between Powers of Regular and General Resolution over Parities

Authors: Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák

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

Abstract

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems.

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Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 23:1-23:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2024.23,
  author =	{Bhattacharya, Sreejata Kishor and Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
  title =	{{Exponential Separation Between Powers of Regular and General Resolution over Parities}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{23:1--23:32},
  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.2024.23},
  URN =		{urn:nbn:de:0030-drops-204191},
  doi =		{10.4230/LIPIcs.CCC.2024.23},
  annote =	{Keywords: Proof Complexity, Regular Reslin, Branching Programs, Lifting}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.108

Two-Source and Affine Non-Malleable Extractors for Small Entropy

Authors: Xin Li and Yan Zhong

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

Non-malleable extractors are generalizations and strengthening of standard randomness extractors, that are resilient to adversarial tampering. Such extractors have wide applications in cryptography and have become important cornerstones in recent breakthroughs of explicit constructions of two-source extractors and affine extractors for small entropy. However, explicit constructions of non-malleable extractors appear to be much harder than standard extractors. Indeed, in the well-studied models of two-source and affine non-malleable extractors, the previous best constructions only work for entropy rate > 2/3 and 1-γ for some small constant γ > 0 respectively by Li (FOCS' 23). In this paper, we present explicit constructions of two-source and affine non-malleable extractors that match the state-of-the-art constructions of standard ones for small entropy. Our main results include: - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k ≥ log^C n and polynomially small error, matching the parameters of standard extractors by Chattopadhyay and Zuckerman (STOC' 16, Annals of Mathematics' 19) and Li (FOCS' 16). - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k = O(log n) and constant error, matching the parameters of standard extractors by Li (FOCS' 23). Our constructions significantly improve previous results, and the parameters (entropy requirement and error) are the best possible without first improving the constructions of standard extractors. In addition, our improved affine non-malleable extractors give strong lower bounds for a certain kind of read-once linear branching programs, recently introduced by Gryaznov, Pudlák, and Talebanfard (CCC' 22) as a generalization of several well studied computational models. These bounds match the previously best-known average-case hardness results given by Chattopadhyay and Liao (CCC' 23) and Li (FOCS' 23), where the branching program size lower bounds are close to optimal, but the explicit functions we use here are different. Our results also suggest a possible deeper connection between non-malleable extractors and standard ones.

Cite as

Xin Li and Yan Zhong. Two-Source and Affine Non-Malleable Extractors for Small Entropy. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 108:1-108:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ICALP.2024.108,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Two-Source and Affine Non-Malleable Extractors for Small Entropy}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{108:1--108:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.108},
  URN =		{urn:nbn:de:0030-drops-202512},
  doi =		{10.4230/LIPIcs.ICALP.2024.108},
  annote =	{Keywords: Randomness Extractors, Non-malleable, Two-source, Affine}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.4

Linear Branching Programs and Directional Affine Extractors

Authors: Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

A natural model of read-once linear branching programs is a branching program where queries are 𝔽₂ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows. - Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then present an explicit construction of such function with good parameters. This strengthens the result of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity decision trees. Directional affine extractors are stronger than the more familiar class of affine extractors. Given the significance of these functions, we expect that our new class of functions might be of independent interest. - Proof complexity. We also consider the proof system Res[⊕], which is an extension of resolution with linear queries, and define the regular variant of Res[⊕]. A refutation of a CNF in this proof system naturally defines a linear branching program solving the corresponding search problem. If a refutation is regular, we prove that the resulting program is read-once. Conversely, we show that a weakly read-once linear BP solving the search problem can be converted to a regular Res[⊕] refutation with constant blow up, where the regularity condition comes from the definition of weakly read-once BPs, thus obtaining the equivalence between these proof systems.

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Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard. Linear Branching Programs and Directional Affine Extractors. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gryaznov_et_al:LIPIcs.CCC.2022.4,
  author =	{Gryaznov, Svyatoslav and Pudl\'{a}k, Pavel and Talebanfard, Navid},
  title =	{{Linear Branching Programs and Directional Affine Extractors}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.4},
  URN =		{urn:nbn:de:0030-drops-165664},
  doi =		{10.4230/LIPIcs.CCC.2022.4},
  annote =	{Keywords: Boolean Functions, Average-Case Lower Bounds, AC0\lbrack2\rbrack, Affine Dispersers, Affine Extractors}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2022.72

A Variant of the VC-Dimension with Applications to Depth-3 Circuits

Authors: Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

We introduce the following variant of the VC-dimension. Given S ⊆ {0,1}ⁿ and a positive integer d, we define 𝕌_d(S) to be the size of the largest subset I ⊆ [n] such that the projection of S on every subset of I of size d is the d-dimensional cube. We show that determining the largest cardinality of a set with a given 𝕌_d dimension is equivalent to a Turán-type problem related to the total number of cliques in a d-uniform hypergraph. This allows us to beat the Sauer-Shelah lemma for this notion of dimension. We use this to obtain several results on Σ₃^k-circuits, i.e., depth-3 circuits with top gate OR and bottom fan-in at most k: - Tight relationship between the number of satisfying assignments of a 2-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). - Improved Σ₃³-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. - We make progress towards settling the Σ₃² complexity of the inner product function and all degree-2 polynomials over 𝔽₂ in general. The question of determining the Σ₃³ complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).

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Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard. A Variant of the VC-Dimension with Applications to Depth-3 Circuits. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 72:1-72:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{frankl_et_al:LIPIcs.ITCS.2022.72,
  author =	{Frankl, Peter and Gryaznov, Svyatoslav and Talebanfard, Navid},
  title =	{{A Variant of the VC-Dimension with Applications to Depth-3 Circuits}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{72:1--72:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.72},
  URN =		{urn:nbn:de:0030-drops-156680},
  doi =		{10.4230/LIPIcs.ITCS.2022.72},
  annote =	{Keywords: VC-dimension, Hypergraph, Clique, Affine Disperser, Circuit}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.3

Proof Complexity of Natural Formulas via Communication Arguments

Authors: Dmitry Itsykson and Artur Riazanov

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract

A canonical communication problem Search(φ) is defined for every unsatisfiable CNF φ: an assignment to the variables of φ is partitioned among the communicating parties, they are to find a clause of φ falsified by this assignment. Lower bounds on the randomized k-party communication complexity of Search(φ) in the number-on-forehead (NOF) model imply tree-size lower bounds, rank lower bounds, and size-space tradeoffs for the formula φ in the semantic proof system T^{cc}(k,c) that operates with proof lines that can be computed by k-party randomized communication protocol using at most c bits of communication [Göös and Pitassi, 2014]. All known lower bounds on Search(φ) (e.g. [Beame et al., 2007; Göös and Pitassi, 2014; Russell Impagliazzo et al., 1994]) are realized on ad-hoc formulas φ (i.e. they were introduced specifically for these lower bounds). We introduce a new communication complexity approach that allows establishing proof complexity lower bounds for natural formulas. First, we demonstrate our approach for two-party communication and apply it to the proof system Res(⊕) that operates with disjunctions of linear equalities over 𝔽₂ [Dmitry Itsykson and Dmitry Sokolov, 2014]. Let a formula PM_G encode that a graph G has a perfect matching. If G has an odd number of vertices, then PM_G has a tree-like Res(⊕)-refutation of a polynomial-size [Dmitry Itsykson and Dmitry Sokolov, 2014]. It was unknown whether this is the case for graphs with an even number of vertices. Using our approach we resolve this question and show a lower bound 2^{Ω(n)} on size of tree-like Res(⊕)-refutations of PM_{K_{n+2,n}}. Then we apply our approach for k-party communication complexity in the NOF model and obtain a Ω(1/k 2^{n/2k - 3k/2}) lower bound on the randomized k-party communication complexity of Search(BPHP^{M}_{2ⁿ}) w.r.t. to some natural partition of the variables, where BPHP^{M}_{2ⁿ} is the bit pigeonhole principle and M = 2ⁿ+2^{n(1-1/k)}. In particular, our result implies that the bit pigeonhole requires exponential tree-like Th(k) proofs, where Th(k) is the semantic proof system operating with polynomial inequalities of degree at most k and k = 𝒪(log^{1-ε} n) for some ε > 0. We also show that BPHP^{2ⁿ+1}_{2ⁿ} superpolynomially separates tree-like Th(log^{1-ε} m) from tree-like Th(log m), where m is the number of variables in the refuted formula.

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Dmitry Itsykson and Artur Riazanov. Proof Complexity of Natural Formulas via Communication Arguments. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 3:1-3:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{itsykson_et_al:LIPIcs.CCC.2021.3,
  author =	{Itsykson, Dmitry and Riazanov, Artur},
  title =	{{Proof Complexity of Natural Formulas via Communication Arguments}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{3:1--3:34},
  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/entities/document/10.4230/LIPIcs.CCC.2021.3},
  URN =		{urn:nbn:de:0030-drops-142773},
  doi =		{10.4230/LIPIcs.CCC.2021.3},
  annote =	{Keywords: bit pigeonhole principle, disjointness, multiparty communication complexity, perfect matching, proof complexity, randomized communication complexity, Resolution over linear equations, tree-like proofs}
}

7 Search Results for "Gryaznov, Svyatoslav"

Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

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Local Enumeration and Majority Lower Bounds

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Exponential Separation Between Powers of Regular and General Resolution over Parities

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Two-Source and Affine Non-Malleable Extractors for Small Entropy

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Linear Branching Programs and Directional Affine Extractors

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A Variant of the VC-Dimension with Applications to Depth-3 Circuits

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Proof Complexity of Natural Formulas via Communication Arguments

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