Article

Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors

Authors:

Ronen Shaltiel,

Avi WigdersonAuthors Info & Claims

STOC '05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing

Pages 1 - 10

https://doi.org/10.1145/1060590.1060592

Published: 22 May 2005 Publication History

Abstract

A distribution X over binary strings of length n has min-entropy k if every string has probability at most 2^-k in X. We say that X is a δ-source if its rate k⁄n is at least δ.We give the following new explicit instructions (namely, poly(n)- time computable functions) of deterministicextractors, dispersers and related objects. All work for any fixed rate δ>0. No previous explicit construction was known for either of these, for any δ‹1⁄2. The first two constitute major progress to very long-standing open problems.

Bipartite Ramsey f₁: (0,1)ⁿ)² →0,1, such that for any two independent δ-sources X₁, X₂ we have f₁(X₁,X₂) = 0,1 This implies a new explicit construction of 2N-vertex bipartite graphs where no induced N^δ by N^δ subgraph is complete or empty.

Multiple source extraction f₂: (0,1n)³→0,1 such that for any three independent δ-sources X₁,X₂,X₃ we have that f₂(X₁,X₂,X₃) is (o(1)-close to being) an unbiased random bit.

Constant seed condenser² f₃: ⁿ →(0,1^m)^c, such that for any δ-source X, one of the c output distributions f₃(X)_i, is a 0.9-source over 0,1^m. Here c is a constant depending only on δ.

Subspace Ramsey f4: 0,1ⁿ→0,1 such that for any affine-δ-source³ X we have f₄(X)= 0,1.

The constructions are quite involved and use as building blocks other new and known gadgets. But we can point out two important themes which recur in these constructions. One is that gadgets which were designed to work with independent inputs, sometimes perform well enough with correlated, high entropy inputs. The second is using the input to (introspectively) find high entropy regions within itself.

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Cited By

Gur TShinkar I(2020)An Entropy Lower Bound for Non-Malleable ExtractorsIEEE Transactions on Information Theory10.1109/TIT.2019.294689666:5(2904-2911)Online publication date: May-2020
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Dodis YFarshim PMazaheri STessaro S(2020)Towards Defeating Backdoored Random Oracles: Indifferentiability with Bounded AdaptivityTheory of Cryptography10.1007/978-3-030-64381-2_9(241-273)Online publication date: 9-Dec-2020
https://doi.org/10.1007/978-3-030-64381-2_9
Li XHatami HMcKenzie PKing V(2017)Improved non-malleable extractors, non-malleable codes and independent source extractorsProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055486(1144-1156)Online publication date: 19-Jun-2017
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Index Terms

Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics

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Reviews

Reviewer: Leo P. Storme

A distribution X over binary strings of length n has min-entropy k if every string has a probability of at most 2-k in X. This distribution X is a Δ-source if its rate k/n is at least Δ. Randomness extraction is the problem of distilling the entropy present in weak random sources into a useful (nearly) uniform distribution. Dispersers are closely related to extractors. Intuitively, a condenser is a function whose output distribution is denser (has a higher entropy rate) than its input distribution. In this paper, explicit constructions, more precisely, poly(n)-time computable functions of deterministic extractors and dispersers, and related objects are given. They all work for any fixed rate larger than zero. This is one of the nice features of the results, since no previous explicit constructions were known for either of these for rates smaller than 1/2. Online Computing Reviews Service

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cover image ACM Conferences

STOC '05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing

May 2005

778 pages

ISBN:1581139608

DOI:10.1145/1060590

General Chair:
Hal Gabow
University of Colorado
,
Program Chair:
Ronald Fagin
IBM Almaden Research Center

Copyright © 2005 ACM.

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Published: 22 May 2005

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Cited By

Gur TShinkar I(2020)An Entropy Lower Bound for Non-Malleable ExtractorsIEEE Transactions on Information Theory10.1109/TIT.2019.294689666:5(2904-2911)Online publication date: May-2020
https://doi.org/10.1109/TIT.2019.2946896
Dodis YFarshim PMazaheri STessaro S(2020)Towards Defeating Backdoored Random Oracles: Indifferentiability with Bounded AdaptivityTheory of Cryptography10.1007/978-3-030-64381-2_9(241-273)Online publication date: 9-Dec-2020
https://doi.org/10.1007/978-3-030-64381-2_9
Li XHatami HMcKenzie PKing V(2017)Improved non-malleable extractors, non-malleable codes and independent source extractorsProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055486(1144-1156)Online publication date: 19-Jun-2017
https://dl.acm.org/doi/10.1145/3055399.3055486
Cohen G(2016)Non-malleable extractorsProceedings of the 31st Conference on Computational Complexity10.5555/2982445.2982453(1-29)Online publication date: 29-May-2016
https://dl.acm.org/doi/10.5555/2982445.2982453
Cohen GShinkar ISudan M(2016)The Complexity of DNF of ParitiesProceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science10.1145/2840728.2840734(47-58)Online publication date: 14-Jan-2016
https://dl.acm.org/doi/10.1145/2840728.2840734
Cohen G(2016)Making the Most of Advice: New Correlation Breakers and Their Applications2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.28(188-196)Online publication date: Oct-2016
https://doi.org/10.1109/FOCS.2016.28
Cohen GSchulman L(2016)Extractors for Near Logarithmic Min-Entropy2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.27(178-187)Online publication date: Oct-2016
https://doi.org/10.1109/FOCS.2016.27
Li X(2015)Three-Source Extractors for Polylogarithmic Min-EntropyProceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2015.58(863-882)Online publication date: 17-Oct-2015
https://dl.acm.org/doi/10.1109/FOCS.2015.58
Cohen G(2015)Local Correlation Breakers and Applications to Three-Source Extractors and MergersProceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2015.57(845-862)Online publication date: 17-Oct-2015
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Cohen GShinkar I(2015)Zero-Fixing Extractors for Sub-Logarithmic EntropyAutomata, Languages, and Programming10.1007/978-3-662-47672-7_28(343-354)Online publication date: 20-Jun-2015
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