9 Search Results for "Guo, Siyao"

Document
DOI: 10.4230/LIPIcs.ITC.2024.9
Time-Space Tradeoffs for Finding Multi-Collisions in Merkle-Damgård Hash Functions

Authors: Akshima

Published in: LIPIcs, Volume 304, 5th Conference on Information-Theoretic Cryptography (ITC 2024)

Abstract
We analyze the multi-collision resistance of Merkle-Damgård hash function construction in the auxiliary input random oracle model. Finding multi-collisions or m-way collisions, for some parameter m, in a hash function consists of m distinct input that have the same output under the hash function. This is a natural generalization of the collision finding problem in hash functions, which is basically finding 2-way collisions. Hardness of finding collisions, or collision resistance, is an important security assumption in cryptography. While the time-space trade-offs for collision resistance of hash functions has received considerable attention, this is the first work that studies time-space trade-offs for the multi-collision resistance property of hash functions based on the popular and widely used Merkle-Damgård (MD) constructions. In this work, we study how the advantage of finding m-way collisions depends on the parameter m. We believe understanding whether multi-collision resistance is a strictly easier property than collision resistance is a fundamental problem and our work facilitates this for adversaries with auxiliary information against MD based hash functions. Furthermore, in this work we study how the advantage varies with the bound on length of the m colliding inputs. Prior works [Akshima et al., 2020; Ashrujit Ghoshal and Ilan Komargodski, 2022; Akshima et al., 2022] have shown that finding "longer" collisions with auxiliary input in MD based hash functions becomes easier. More precisely, the advantage of finding collisions linearly depends on the bound on the length of colliding inputs. In this work, we show similar dependence for m-way collision finding, for any m ≥ 2. We show a simple attack for finding 1-block m-way collisions which achieves an advantage of Ω̃(S/mN). For 2 ≤ B < log m, we give the best known attack for finding B-blocks m-way collision which achieves an advantage of Ω̃(ST/m^{1/(B-1)}N) when m^{1/(B-1)}-way collisions exist on every salt. For B > log m, our attack achieves an advantage of Ω̃(STB/N) which is optimal when SB ≥ T and ST² ≤ N. The main results of this work is showing that our attacks are optimal for B = 1 and B = 2. This implies that in the auxiliary-input random oracle model, the advantage decreases by a multiplicative factor of m for finding 1-block and 2-block m-way collisions (compared to collision finding) in Merkle-Damgård based hash functions.
Cite as

Akshima. Time-Space Tradeoffs for Finding Multi-Collisions in Merkle-Damgård Hash Functions. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{akshima:LIPIcs.ITC.2024.9,
  author =	{Akshima},
  title =	{{Time-Space Tradeoffs for Finding Multi-Collisions in Merkle-Damg\r{a}rd Hash Functions}},
  booktitle =	{5th Conference on Information-Theoretic Cryptography (ITC 2024)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-333-1},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{304},
  editor =	{Aggarwal, Divesh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2024.9},
  URN =		{urn:nbn:de:0030-drops-205171},
  doi =		{10.4230/LIPIcs.ITC.2024.9},
  annote =	{Keywords: Collision, hash functions, multi-collisions, Merkle-Damg\r{a}rd, pre-computation, auxiliary input}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.10
The (Im)possibility of Simple Search-To-Decision Reductions for Approximation Problems

Authors: Alexander Golovnev, Siyao Guo, Spencer Peters, and Noah Stephens-Davidowitz

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
We study the question of when an approximate search optimization problem is harder than the associated decision problem. Specifically, we study a natural and quite general model of black-box search-to-decision reductions, which we call branch-and-bound reductions (in analogy with branch-and-bound algorithms). In this model, an algorithm attempts to minimize (or maximize) a function f: D → ℝ_{≥ 0} by making oracle queries to h_f : 𝒮 → ℝ_{≥ 0} satisfying min_{x ∈ S} f(x) ≤ h_f(S) ≤ γ ⋅ min_{x ∈ S} f(x) (*) for some γ ≥ 1 and any subset S in some allowed class of subsets 𝒮 of the domain D. (When the goal is to maximize f, h_f instead yields an approximation to the maximal value of f over S.) We show tight upper and lower bounds on the number of queries q needed to find even a γ'-approximate minimizer (or maximizer) for quite large γ' in a number of interesting settings, as follows. - For arbitrary functions f : {0,1}ⁿ → ℝ_{≥ 0}, where 𝒮 contains all subsets of the domain, we show that no branch-and-bound reduction can achieve γ' ≲ γ^{n/log q}, while a simple greedy approach achieves essentially γ^{n/log q}. - For a large class of MAX-CSPs, where 𝒮 := {S_w} contains each set of assignments to the variables induced by a partial assignment w, we show that no branch-and-bound reduction can do significantly better than essentially a random guess, even when the oracle h_f guarantees an approximation factor of γ ≈ 1+√{log(q)/n}. - For the Traveling Salesperson Problem (TSP), where 𝒮 := {S_p} contains each set of tours extending a path p, we show that no branch-and-bound reduction can achieve γ' ≲ (γ-1) n/log q. We also prove a nearly matching upper bound in our model. These results show an oracle model in which approximate search and decision are strongly separated. (In particular, our result for TSP can be viewed as a negative answer to a question posed by Bellare and Goldwasser (SIAM J. Comput. 1994), though only in an oracle model.) We also note two alternative interpretations of our results. First, if we view h_f as a data structure, then our results unconditionally rule out black-box search-to-decision reductions for certain data structure problems. Second, if we view h_f as an efficiently computable heuristic, then our results show that any reasonably efficient branch-and-bound algorithm requires more guarantees from its heuristic than simply Eq. (*). Behind our results is a "useless oracle lemma," which allows us to argue that under certain conditions the oracle h_f is "useless," and which might be of independent interest. See also the full version [Alexander Golovnev et al., 2022].
Cite as

Alexander Golovnev, Siyao Guo, Spencer Peters, and Noah Stephens-Davidowitz. The (Im)possibility of Simple Search-To-Decision Reductions for Approximation Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{golovnev_et_al:LIPIcs.APPROX/RANDOM.2023.10,
  author =	{Golovnev, Alexander and Guo, Siyao and Peters, Spencer and Stephens-Davidowitz, Noah},
  title =	{{The (Im)possibility of Simple Search-To-Decision Reductions for Approximation Problems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.10},
  URN =		{urn:nbn:de:0030-drops-188351},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.10},
  annote =	{Keywords: search-to-decision reductions, oracles, constraint satisfaction, traveling salesman, discrete optimization}
}
Document
DOI: 10.4230/LIPIcs.ITC.2022.6
On the Distributed Discrete Logarithm Problem with Preprocessing

Authors: Pavel Hubáček, Ľubica Jančová, and Veronika Králová

Published in: LIPIcs, Volume 230, 3rd Conference on Information-Theoretic Cryptography (ITC 2022)

Abstract
Protocols solving the Distributed Discrete Logarithm (DDLog) problem are a core component of many recent constructions of group-based homomorphic secret sharing schemes. On a high-level, these protocols enable two parties to transform multiplicative shares of a secret into additive share locally without any communication. Due to their important applications, various generic optimized DDLog protocols were proposed in the literature, culminating in the asymptotically optimal generic protocol of Dinur, Keller, and Klein (J. Cryptol. 2020) solving DDLog in time T with error probability O(W/T²) when the magnitude of the secret is bounded by W. Given that DDLog is solved repeatedly with respect to a fixed group in its applications, a natural approach for improving the efficiency of DDLog protocols could be via leveraging some precomputed group-specific advice. To understand the limitations of this approach, we revisit the distributed discrete logarithm problem in the preprocessing model and study the possible time-space trade-offs for DDLog in the generic group model. As our main result, we show that, in a group of size N, any generic DDLog protocol for secrets of magnitude W with parties running in time T using precomputed group-specific advice of size S has success probability ε = O (T²/W + max{S,log W} ⋅ T²/N) . Thus, assuming N ≥ W log W, we get a lower bound ST² = Ω(ε N) on the time-space trade-off for DDLog protocols using large advice of size S = Ω(N/W). Interestingly, for DDLog protocols using small advice of size S = O(N/W), we get a lower bound T² = Ω(ε W) on the running time, which, in the constant-error regime, asymptotically matches the running time of the DDLog protocol without any advice of Dinur et al. (J. Cryptol. 2020). In other words, we show that generic DDLog protocols achieving constant success probability do not benefit from any advice of size S = O(N/W) in the online phase of the DDLog problem.
Cite as

Pavel Hubáček, Ľubica Jančová, and Veronika Králová. On the Distributed Discrete Logarithm Problem with Preprocessing. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hubacek_et_al:LIPIcs.ITC.2022.6,
  author =	{Hub\'{a}\v{c}ek, Pavel and Jan\v{c}ov\'{a}, \v{L}ubica and Kr\'{a}lov\'{a}, Veronika},
  title =	{{On the Distributed Discrete Logarithm Problem with Preprocessing}},
  booktitle =	{3rd Conference on Information-Theoretic Cryptography (ITC 2022)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-238-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{230},
  editor =	{Dachman-Soled, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2022.6},
  URN =		{urn:nbn:de:0030-drops-164847},
  doi =		{10.4230/LIPIcs.ITC.2022.6},
  annote =	{Keywords: Distributed discrete logarithm problem, preprocessing, generic group model}
}
Document
DOI: 10.4230/LIPIcs.ITC.2021.14
Online Linear Extractors for Independent Sources

Authors: Yevgeniy Dodis, Siyao Guo, Noah Stephens-Davidowitz, and Zhiye Xie

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

Abstract
In this work, we characterize linear online extractors. In other words, given a matrix A ∈ F₂^{n×n}, we study the convergence of the iterated process S ← AS⊕X, where X∼D is repeatedly sampled independently from some fixed (but unknown) distribution D with (min)-entropy k. Here, we think of S ∈ {0,1}ⁿ as the state of an online extractor, and X ∈ {0,1}ⁿ as its input. As our main result, we show that the state S converges to the uniform distribution for all input distributions D with entropy k > 0 if and only if the matrix A has no non-trivial invariant subspace (i.e., a non-zero subspace V ⊊ F₂ⁿ such that AV ⊆ V). In other words, a matrix A yields a linear online extractor if and only if A has no non-trivial invariant subspace. For example, the linear transformation corresponding to multiplication by a generator of the field F_{2ⁿ} yields a good linear online extractor. Furthermore, for any such matrix convergence takes at most Õ(n²(k+1)/k²) steps. We also study the more general notion of condensing - that is, we ask when this process converges to a distribution with entropy at least l, when the input distribution has entropy at least k. (Extractors corresponding to the special case when l = n.) We show that a matrix gives a good condenser if there are relatively few vectors w ∈ F₂ⁿ such that w, A^Tw, …, (A^T)^{n-k}w are linearly dependent. As an application, we show that the very simple cyclic rotation transformation A(x₁,…, x_n) = (x_n,x₁,…, x_{n-1}) condenses to l = n-1 bits for any k > 1 if n is a prime satisfying a certain simple number-theoretic condition. Our proofs are Fourier-analytic and rely on a novel lemma, which gives a tight bound on the product of certain Fourier coefficients of any entropic distribution.
Cite as

Yevgeniy Dodis, Siyao Guo, Noah Stephens-Davidowitz, and Zhiye Xie. Online Linear Extractors for Independent Sources. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dodis_et_al:LIPIcs.ITC.2021.14,
  author =	{Dodis, Yevgeniy and Guo, Siyao and Stephens-Davidowitz, Noah and Xie, Zhiye},
  title =	{{Online Linear Extractors for Independent Sources}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.14},
  URN =		{urn:nbn:de:0030-drops-143339},
  doi =		{10.4230/LIPIcs.ITC.2021.14},
  annote =	{Keywords: feasibility of randomness extraction, randomness condensers, Fourier analysis}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.1
Extractor Lower Bounds, Revisited

Authors: Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro, and Noah Stephens-Davidowitz

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a "change in quantifiers" over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input sources with sufficient min-entropy, a somewhere extractor only requires that there exists a seed whose output bias is small. More generally, we study what we call probable extractors, which on input a source with sufficient min-entropy guarantee that a large enough fraction of seeds have small enough associated output bias. Such extractors have played a key role in many constructions of pseudorandom objects, though they are often defined implicitly and have not been studied extensively. Prior known techniques fail to yield good seed length lower bounds when applied to the variants above. Our novel approach yields significantly improved lower bounds for somewhere and probable extractors. To complement this, we construct a somewhere extractor that implies our lower bound for such functions is tight in the high min-entropy regime. Surprisingly, this means that a random function is far from an optimal somewhere extractor in this regime. The techniques that we develop also yield an alternative, simpler proof of the celebrated optimal lower bound for strong extractors originally due to Radhakrishnan and Ta-Shma (SIAM J. Discrete Math., 2000).
Cite as

Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro, and Noah Stephens-Davidowitz. Extractor Lower Bounds, Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aggarwal_et_al:LIPIcs.APPROX/RANDOM.2020.1,
  author =	{Aggarwal, Divesh and Guo, Siyao and Obremski, Maciej and Ribeiro, Jo\~{a}o and Stephens-Davidowitz, Noah},
  title =	{{Extractor Lower Bounds, Revisited}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.1},
  URN =		{urn:nbn:de:0030-drops-126041},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.1},
  annote =	{Keywords: randomness extractors, lower bounds, explicit constructions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.86
On the Complexity of Decomposable Randomized Encodings, Or: How Friendly Can a Garbling-Friendly PRF Be?

Authors: Marshall Ball, Justin Holmgren, Yuval Ishai, Tianren Liu, and Tal Malkin

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

Abstract
Garbling schemes, also known as decomposable randomized encodings (DRE), have found many applications in cryptography. However, despite a large body of work on constructing such schemes, very little is known about their limitations. We initiate a systematic study of the DRE complexity of Boolean functions, obtaining the following main results: - Near-quadratic lower bounds. We use a classical lower bound technique of Nečiporuk [Dokl. Akad. Nauk SSSR '66] to show an Ω(n²/log n) lower bound on the size of any DRE for many explicit Boolean functions. For some natural functions, we obtain a corresponding upper bound, thus settling their DRE complexity up to polylogarithmic factors. Prior to our work, no superlinear lower bounds were known, even for non-explicit functions. - Garbling-friendly PRFs. We show that any exponentially secure PRF has Ω(n²/log n) DRE size, and present a plausible candidate for a "garbling-optimal" PRF that nearly meets this bound. This candidate establishes a barrier for super-quadratic DRE lower bounds via natural proof techniques. In contrast, we show a candidate for a weak PRF with near-exponential security and linear DRE size. Our results establish several qualitative separations, including near-quadratic separations between computational and information-theoretic DRE size of Boolean functions, and between DRE size of weak vs. strong PRFs.
Cite as

Marshall Ball, Justin Holmgren, Yuval Ishai, Tianren Liu, and Tal Malkin. On the Complexity of Decomposable Randomized Encodings, Or: How Friendly Can a Garbling-Friendly PRF Be?. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 86:1-86:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ball_et_al:LIPIcs.ITCS.2020.86,
  author =	{Ball, Marshall and Holmgren, Justin and Ishai, Yuval and Liu, Tianren and Malkin, Tal},
  title =	{{On the Complexity of Decomposable Randomized Encodings, Or: How Friendly Can a Garbling-Friendly PRF Be?}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{86:1--86:22},
  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.2020.86},
  URN =		{urn:nbn:de:0030-drops-117714},
  doi =		{10.4230/LIPIcs.ITCS.2020.86},
  annote =	{Keywords: Randomized Encoding, Private Simultaneous Messages}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.30
Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources

Authors: Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
Let F be a finite alphabet and D be a finite set of distributions over F. A Generalized Santha-Vazirani (GSV) source of type (F, D), introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence (F_1, ..., F_n) in F^n, where F_i is a sample from some distribution d in D whose choice may depend on F_1, ..., F_{i-1}. We show that all GSV source types (F, D) fall into one of three categories: (1) non-extractable; (2) extractable with error n^{-Theta(1)}; (3) extractable with error 2^{-Omega(n)}. We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts one bit with error epsilon from n = poly(1/epsilon) samples in time linear in n. Our algorithm for category (3) sources extracts m bits with error epsilon from n = O(m + log 1/epsilon) samples in time min{O(m2^m * n),n^{O(|F|)}}. We also give algorithms for classifying a GSV source type (F, D): Membership in category (1) can be decided in NP, while membership in category (3) is polynomial-time decidable.
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Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo. Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beigi_et_al:LIPIcs.APPROX-RANDOM.2018.30,
  author =	{Beigi, Salman and Bogdanov, Andrej and Etesami, Omid and Guo, Siyao},
  title =	{{Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.30},
  URN =		{urn:nbn:de:0030-drops-94349},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.30},
  annote =	{Keywords: feasibility of randomness extraction, extractor lower bounds, martingales}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.29
Testing k-Monotonicity

Authors: Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions. Motivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone. Our results include the following: 1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0,1}^d, for k >= 3; 2. We demonstrate a separation between testing and learning on {0,1}^d, for k=\omega(\log d): testing k-monotonicity can be performed with 2^{O(\sqrt d . \log d . \log{1/\eps})} queries, while learning k-monotone functions requires 2^{\Omega(k . \sqrt d .{1/\eps})} queries (Blais et al. (RANDOM 2015)). 3. We present a tolerant test for functions f\colon[n]^d\to \{0,1\}$with complexity independent of n, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques. Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.
Cite as

Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer. Testing k-Monotonicity. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 29:1-29:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{canonne_et_al:LIPIcs.ITCS.2017.29,
  author =	{Canonne, Cl\'{e}ment L. and Grigorescu, Elena and Guo, Siyao and Kumar, Akash and Wimmer, Karl},
  title =	{{Testing k-Monotonicity}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{29:1--29:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.29},
  URN =		{urn:nbn:de:0030-drops-81583},
  doi =		{10.4230/LIPIcs.ITCS.2017.29},
  annote =	{Keywords: Boolean Functions, Learning, Monotonicity, Property Testing}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.850
Negation-Limited Formulas

Authors: Siyao Guo and Ilan Komargodski

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract
We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC'15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.
Cite as

Siyao Guo and Ilan Komargodski. Negation-Limited Formulas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 850-866, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{guo_et_al:LIPIcs.APPROX-RANDOM.2015.850,
  author =	{Guo, Siyao and Komargodski, Ilan},
  title =	{{Negation-Limited Formulas}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{850--866},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.850},
  URN =		{urn:nbn:de:0030-drops-53400},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.850},
  annote =	{Keywords: Negation complexity, De Morgan formulas, Shrinkage}
}
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