Three-Source Extractors for Polylogarithmic Min-Entropy

Li, Xin

doi:10.1109/focs.2015.58

Cited by 29 publications

(15 citation statements)

References 34 publications

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“…The above two kinds of extractors are closely related, and in many cases techniques used for one can also be used to improve the constructions of the other. These connections have been demonstrated in a number of works (e.g., [Li12b,Li13b,Li13a,Li15d,CZ16]).…”

Section: Introductionmentioning

confidence: 84%

Explicit Non-malleable Extractors, Multi-source Extractors, and Almost Optimal Privacy Amplification Protocols

Chattopadhyay

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We make progress in the following three problems: 1. Constructing optimal seeded nonmalleable extractors; 2. Constructing optimal privacy amplification protocols with an active adversary, for any possible security parameter; 3. Constructing extractors for independent weak random sources, when the min-entropy is extremely small (i.e., near logarithmic).For the first two problems, the best known non-malleable extractors by Chattopadhyay, Goyal and Li [CGL16], and by Cohen [Coh16a,Coh16b] all require seed length and min-entropy at least log 2 (1/ε), where ε is the error of the extractor. As a result, the best known explicit privacy amplification protocols with an active adversary, which achieve 2 rounds of communication and optimal entropy loss in [Li15c, CGL16], can only handle security parameter up to s = Ω( √ k), where k is the min-entropy of the shared secret weak random source. For larger s the best known protocol with optimal entropy loss in [Li15c] requires O(s/ √ k) rounds of communication.In this paper we give an explicit non-malleable extractor that only requires seed length and min-entropy log 1+o(1) (n/ε), which also yields a 2-round privacy amplification protocol with optimal entropy loss for security parameter up to s = k 1−α for any constant α > 0. For the third problem, previously the best known extractor which supports the smallest min-entropy due to Li [Li13a], requires min-entropy log 2+δ n and uses O(1/δ) sources, for any constant δ > 0. A very recent result by Cohen and Schulman [CS16] improves this, and constructed explicit extractors that use O(1/δ) sources for min-entropy log 1+δ n, any constant δ > 0. In this paper we further improve their result, and give an explicit extractor that uses O(1) (an absolute constant) sources for min-entropy log 1+o(1) n. The key ingredient in all our constructions is a generalized, and much more efficient version of the independence preserving merger introduced in [CS16], which we call non-malleable independence preserving merger. Our construction of the merger also simplifies that of [CS16], and may be of independent interest.

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Section: Introductionmentioning

confidence: 84%

Explicit Non-malleable Extractors, Multi-source Extractors, and Almost Optimal Privacy Amplification Protocols

Chattopadhyay

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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“…In a recent breakthrough, Li [Li15] gave a construction of an extractor BExt for two n-bit sources, where the first source is a polylog(n)-block-source and the second is a weak-source with min-entropy polylog(n) (see Theorem 4.1). Since our goal is to construct a two-source sub-extractor for outer-entropy polylog(n), a first attempt would be to show that any source X with entropy polylog(n) has a subsource X that is a polylog(n)-block-source.…”

Section: Motivating the Challenge-response Mechanismmentioning

confidence: 99%

Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

Cohen¹

2019

SIAM J. Comput.

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In his 1947 paper that inaugurated the probabilistic method, Erdös [Erd47] proved the existence of 2 log n-Ramsey graphs on n vertices. Matching Erdös' result with a constructive proof is a central problem in combinatorics, that has gained a significant attention in the literature. The state of the art result was obtained in the celebrated paper by Barak, Rao, Shaltiel and Wigderson [Ann. Math'12], who constructed a 2 2 (log log n) 1−α -Ramsey graph, for some small universal constant α > 0. In this work, we significantly improve the result of Barak et al. and construct 2 (log log n) c -Ramsey graphs, for some universal constant c. In the language of theoretical computer science, our work resolves the problem of explicitly constructing two-source dispersers for polylogarithmic entropy.

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“…Chattopadhyay and Zuckerman reduce the problem of computing two-source extractors to that of constructing an explicit n 1−δ -resilient function for some constant δ > 0 with the following stronger property; this reduction was also implicit in [Li15]. A distribution D on {0, 1} n is t-wise independent if for X ← D, and all I ⊆ [n] with |I| ≤ t, the projection of X onto the coordinates in I, X I , is uniformly distributed over {0, 1}…”

Section: Two-source Extractorsmentioning

confidence: 99%

Explicit Resilient Functions Matching Ajtai-Linial

Meka¹

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A Boolean function on n variables is q-resilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining n − q variables; in other words, no coalition of at most q variables has significant influence on the function. Resilient functions have been extensively studied with a variety of applications in cryptography, distributed computing, and pseudorandomness. The best known resilient function on n variables due to Ajtai and Linial [AL93] has the property that only sets of size Ω(n/(log 2 n)) can have influence bounded away from zero. However, the construction of Ajtai and Linial is by the probabilistic method and does not give an efficiently computable function.We construct an explicit monotone depth three almostbalanced Boolean function on n bits that is Ω(n/(log 2 n))-resilient matching the bounds of Ajtai and Linial. The best previous explicit constructions of Meka [Mek09] (which only gives a logarithmic depth function), and Chattopadhyay and Zuckerman [CZ15] were only (n 1−β )-resilient for any constant 0 < β < 1. Our construction and analysis are motivated by (and simplifies parts of) the recent breakthrough of [CZ15] giving explicit two-sources extractors for polylogarithmic min-entropy; a key ingredient in their result was the construction of explicit constant-depth resilient functions.An important ingredient in our construction is a new randomness-optimal oblivious sampler that preserves moment generating functions of sums of variables and could be useful elsewhere.

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Three-Source Extractors for Polylogarithmic Min-Entropy

Cited by 29 publications

References 34 publications

Explicit Non-malleable Extractors, Multi-source Extractors, and Almost Optimal Privacy Amplification Protocols

Explicit Non-malleable Extractors, Multi-source Extractors, and Almost Optimal Privacy Amplification Protocols

Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

Explicit Resilient Functions Matching Ajtai-Linial

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