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Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

Dean Doron, Pooya Hatami, William M. Hoza, Michael Wagner
2019 Computational Complexity Conference  
We give an explicit pseudorandom generator (PRG) for read-once AC 0 , i.e., constant-depth read-once formulas over the basis {∧, ∨, ¬} with unbounded fan-in.  ...  Previously, PRGs with near-optimal seed length were known only for the depth-2 case [22] .  ...  Acknowledgements We thank David Zuckerman for very helpful discussions.  ... 
doi:10.4230/lipics.ccc.2019.16 dblp:conf/coco/DoronHH19 fatcat:laa67gulbbgophzbroctb46lsq

Algebraic Methods in Computational Complexity (Dagstuhl Seminar 18391)

Markus Bläser, Valentine Kabanets, Jacobo Torán, Christopher Umans, Michael Wagner
2019 Dagstuhl Reports  
The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some  ...  would lead to a general result.  ...  Pseudorandom Generators for Read-Once Branching Programs, in any Order Michael A.  ... 
doi:10.4230/dagrep.8.9.133 dblp:journals/dagstuhl-reports/BlaserKTU18 fatcat:bqddvcedazgqngwix6ptk7syuq

Log-Seed Pseudorandom Generators via Iterated Restrictions

Dean Doron, Pooya Hatami, William M. Hoza, Shubhangi Saraf
2020 Computational Complexity Conference  
Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)).  ...  Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019].  ...  Acknowledgements We thank David Zuckerman for very helpful discussions.  ... 
doi:10.4230/lipics.ccc.2020.6 dblp:conf/coco/DoronHH20 fatcat:ymkmdzvqy5dzfiugl2fg4zneym

Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)

Manindra Agrawal, Thomas Thierauf, Christopher Umans, Marc Herbstritt
2013 Dagstuhl Reports  
Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically.  ...  For de Morgan formulas, seed length s 1/3+o(1) ; 2. For formulas over an arbitrary basis, seed length s 1/2+o(1) ; 3. For read-once de Morgan formulas, seed length s 0.234... ; 4.  ...  by Agrawal and Vinay to show a general non-trivial reduction to depth 4.  ... 
doi:10.4230/dagrep.2.10.60 dblp:journals/dagstuhl-reports/AgrawalTU12 fatcat:dg7ithf6xfgadkzwkxjtjhy7ge

How to Find Water in the Ocean: A Survey on Quantified Derandomization [article]

Roei Tell
2021 Electronic colloquium on computational complexity  
How small does this probability need to be in order for the problem's complexity to be affected?  ...  gradual progress towards solving the general case.  ...  I thank Ryan Williams for encouraging me to write the survey, for pointing out that Theorem 2.6 is a strict generalization of his result [Wil13] , and for providing good writing advice.  ... 
dblp:journals/eccc/Tell21 fatcat:dntqqo67ercrzjkog66xz4iyxu

The exact complexity of pseudorandom functions and the black-box natural proof barrier for bootstrapping results in computational complexity

Zhiyuan Fan, Jiatu Li, Tianqi Yang
2022 Symposium on the Theory of Computing  
We also present an 𝑛 1+Ω (𝑐 −𝑑 ) wire complexity lower bound against depth-𝑑 TC 0 circuits for some 𝑐 > 1.61.  ...  This paper provides answers to this problem on pseudorandom functions (PRFs).  ...  Oliveira for organizing a virtual seminar and valuable comments. We are thankful to Tianyi Zhang for addressing a typo in an earlier draft and Yiding Zhang for his help in improving some writing.  ... 
doi:10.1145/3519935.3520010 dblp:conf/stoc/FanL022 fatcat:axkbbap6tbf3bd7456k7lgsahq

Efficient Multiparty Protocols via Log-Depth Threshold Formulae [chapter]

Gil Cohen, Ivan Bjerre Damgård, Yuval Ishai, Jonas Kölker, Peter Bro Miltersen, Ran Raz, Ron D. Rothblum
2013 Lecture Notes in Computer Science  
Cryptology, 2000) with constructions of logarithmic-depth formulae which compute threshold functions using only constant fan-in threshold gates.  ...  We put forward a new approach for the design of efficient multiparty protocols: 1.  ...  The state of the art pseudorandom generator for read once branching programs has seed length O(log 2 n) [Nis92, INW94] .  ... 
doi:10.1007/978-3-642-40084-1_11 fatcat:a7lwyvx6d5a2xbujlmoe6b7npe

DNF Sparsification and a Faster Deterministic Counting Algorithm [article]

Parikshit Gopala, Raghu Meka, Omer Reingold
2012 arXiv   pre-print
Given a formula on n variables with poly(n) terms, we give a deterministic n^Õ((n)) time algorithm that computes an additive ϵ approximation to the fraction of satisfying assignments of f for ϵ = 1/( n  ...  It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified.  ...  We thank Rocco for drawing our attention to Friedgut's theorem in this context.  ... 
arXiv:1205.3534v1 fatcat:e23ypu6yafgzni35gvz6eqzqgi

Better Pseudorandom Generators from Milder Pseudorandom Restrictions

Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, Salil Vadhan
2012 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science  
We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs, all of which achieve near-optimal seed-length  ...  We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions.  ...  We will show that, for read-once CNFs, such a pseudorandom restriction can be generated using O(log(m/ε)) random bits.  ... 
doi:10.1109/focs.2012.77 dblp:conf/focs/GopalanMRTV12 fatcat:4juxxydijne27bthnzi4crifg4

Better Pseudorandom Generators from Milder Pseudorandom Restrictions [article]

Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, Salil Vadhan
2012 arXiv   pre-print
We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs, all of which achieve near-optimal seed-length  ...  We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions.  ...  We will show that, for read-once CNFs, such a pseudorandom restriction can be generated using O(log(m/ε)) random bits.  ... 
arXiv:1210.0049v1 fatcat:zqsjkn2tpfarpm3pncll2urtzm

Optimal Cryptographic Hardness of Learning Monotone Functions [chapter]

Dana Dachman-Soled, Homin K. Lee, Tal Malkin, Rocco A. Servedio, Andrew Wan, Hoeteck Wee
2008 Lecture Notes in Computer Science  
Lower bounds of the type we establish have previously only been known for non-monotone functions.  ...  Some of our results show cryptographic hardness of learning polynomial-size monotone circuits to accuracy only slightly greater than 1/2 + 1/ √ n; this accuracy bound is close to optimal by known positive  ...  We prove near-optimal hardness for learning polynomialsize monotone circuits: Theorem 1 (informal).  ... 
doi:10.1007/978-3-540-70575-8_4 fatcat:v6zgngwvpzb27jhxalzfytr6d4

Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, Christopher Umans, Marc Herbstritt
2017 Dagstuhl Reports  
The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some  ...  would lead to a general result.  ...  We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy.  ... 
doi:10.4230/dagrep.6.10.13 dblp:journals/dagstuhl-reports/KabanetsTTU16 fatcat:mxexvi3xmngw7iwfc2bodtsd3y

Circuit Depth Reductions [article]

Alexander Golovnev and Alexander S. Kulikov and R. Ryan Williams
2020 arXiv   pre-print
for depth-3 circuits with constant bottom fan-in.  ...  We show that improvements of the following pseudorandom constructions imply new circuit lower bounds: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness  ...  most t, and there exists a read-once formula F ′ of size k such that F ′ (T 1 (x), ..., T k (x)) = F (x) for all x ∈ {0, 1} n .  ... 
arXiv:1811.04828v4 fatcat:r2jkgigmqncctbxhfahk2meug4

Pseudorandomness

Salil P. Vadhan
2012 Foundations and Trends® in Theoretical Computer Science  
graphs, randomness extractors, list-decodable error-correcting codes, samplers, and pseudorandom generators.  ...  This is a survey of pseudorandomness, the theory of efficiently generating objects that "look random" despite being constructed using little or no randomness.  ...  I am indebted to Oded, Shafi, Madhu, Avi, Luca, and Omer for all the insights and research experiences they have shared with me.  ... 
doi:10.1561/0400000010 fatcat:2xv2ssm7lbhnjktg6l3u5o5kfu

Beyond Natural Proofs: Hardness Magnification and Locality [article]

Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Jan Pich, Ninad Rajgopal, Rahul Santhanam
2019 arXiv   pre-print
for magnification.  ...  Can we adapt known lower bound techniques to establish the desired lower bound for Q?  ...  Acknowledgements Part of this work was completed while some of the authors were visiting the Simons Institute for the Theory of Computing. We are grateful to the Simons Institute for their support.  ... 
arXiv:1911.08297v1 fatcat:zfitgvmtzngl7elbcqyfvr5m6i
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