Explicit two-source extractors and resilient functions

Chattopadhyay, Eshan; Zuckerman, David

doi:10.1145/2897518.2897528

Cited by 81 publications

(86 citation statements)

References 51 publications

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“…The proof of Theorem 2.1 is similar to the arguments in [CZ15] and relies on Janson's inequality. However, our argument is more subtle as we need to handle (d, k, δ)-designs and not just d-designs as is done there.…”

Section: Overview Of Constructionmentioning

confidence: 90%

“…One of the main building blocks of their work is an efficiently computable resilient function with a stronger guarantee described below. We noticeably simplify the construction and analysis of [CZ15] and obtain better quantitative bounds. We explain these next starting with the definitions of extractors.…”

Section: Two-source Extractorsmentioning

confidence: 99%

“…−Ω(k) for all k ≥ 2 log n (even outputting Ω(k) bits; as in [CZ15] we only focus on extracting one bit in this work). Constructing such functions explicitly, as is required in most applications, is much harder and has been studied extensively.…”

Section: Two-source Extractorsmentioning

confidence: 99%

“…Two-source extractors and resilient functions Along with the above quantitative improvements, our construction and analysis simplify [CZ15]:…”

Section: Two-source Extractorsmentioning

confidence: 99%

“…To analyze such functions, we first state two abstract properties that allow us to analyze the bias as well as influences of such functions; we then design partitions that satisfy the properties. The properties we define are motivated by [CZ15] and abstracting them in this way allows us to give a modular analysis of the construction. The partitions themselves will be designed using the sampler we construct in Theorem 1.5 which forms the core of our analysis and construction.…”

Section: Overview Of Constructionmentioning

confidence: 99%

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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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Section: Overview Of Constructionmentioning

confidence: 90%

Section: Two-source Extractorsmentioning

confidence: 99%

Section: Two-source Extractorsmentioning

confidence: 99%

“…Two-source extractors and resilient functions Along with the above quantitative improvements, our construction and analysis simplify [CZ15]:…”

Section: Two-source Extractorsmentioning

confidence: 99%

Section: Overview Of Constructionmentioning

confidence: 99%