Simple extractors for all min-entropies and a new pseudo-random generator

Shaltiel, Ronen; Umans, Christopher

doi:10.1109/sfcs.2001.959941

Cited by 78 publications

(99 citation statements)

References 34 publications

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“…Miltersen & Vinodchandran (1999) prove that if NE ∩ coNE ⊆ NTIME(2 βn )/2 βn for some constant β then AM = NP. This high-end result was extended in Shaltiel & Umans (2001) and Umans (2002) to the low-end setting.…”

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confidence: 82%

“…Today, after a long line of research (Babai et al 1993;Blum & Micali 1984;Impagliazzo et al 1999Impagliazzo et al , 2000Impagliazzo & Wigderson 1997;Klivans & van Melkebeek 1999;Miltersen & Vinodchandran 1999;Nisan & Wigderson 1994;Shaltiel & Umans 2001;Sudan et al 1999;Umans 2002;Yao 1982), we have powerful and elegant constructions of low-end and high-end generators that derandomize BPP, MA and AM using "necessary assumptions" (i.e., assumptions that are implied by the existence of pseudo-random generators against nonuniform circuits). The reader is referred to a recent survey paper on derandomization for more details (Kabanets 2002).…”

Section: Introductionmentioning

confidence: 99%

“…• Nonuniform hardness vs. randomness tradeoffs for AM were given in Klivans & van Melkebeek (1999);Miltersen & Vinodchandran (1999); Shaltiel & Umans (2001). Miltersen & Vinodchandran (1999) prove that if NE ∩ coNE ⊆ NTIME(2 βn )/2 βn for some constant β then AM = NP.…”

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confidence: 99%

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Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

2003

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Abstract. Impagliazzo and Wigderson proved a uniform hardness vs. randomness "gap theorem" for BPP. We show an analogous result for AM: Either Arthur-Merlin protocols are very strong and everything in E = DTIME(2 O(n) ) can be proved to a subexponential time verifier, or else Arthur-Merlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm such that it is infeasible to come up with inputs on which the algorithm fails. We also show that if Arthur-Merlin protocols are not very strong (in the sense explained above) then AM ∩ coAM = NP ∩ coNP. Our technique combines the nonuniform hardness versus randomness tradeoff of Miltersen and Vinodchandran with "instance checking". A key ingredient in our proof is identifying a novel "resilience" property of hardness vs. randomness tradeoffs.

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confidence: 82%

Section: Introductionmentioning

confidence: 99%

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Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

2003

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“…In the above argument, the Nisan-Wigderson construction may be replaced by the recent pseudorandom generators or reconstructive extractors based on multivariate polynomials (Shaltiel & Umans 2001;Ta-Shma et al 2001b). However, although the latter lead to optimal hardness versus randomness tradeoffs in some sense (Umans 2003), they yield worse parameters than the NisanWigderson construction in our context.…”

Section: Our Techniquementioning

confidence: 99%

“…The technique of providing approximations to the average number of positive NP queries to limit Merlin's ability to cheat has been exploited before, cc 14 (2005) Language compression and pseudorandom generators 245 e.g., in the context of random selfreducibility (Feigenbaum & Fortnow 1993) and more recently in hardness versus randomness tradeoffs for nondeterministic circuits (Shaltiel & Umans 2001). …”

Section: Language Compression By Nondeterminism and Randomnessmentioning

confidence: 99%

Language compression and pseudorandom generators

2005

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Abstract. The language compression problem asks for succinct descriptions of the strings in a language A such that the strings can be efficiently recovered from their description when given a membership oracle for A. We study randomized and nondeterministic decompression schemes and investigate how close we can get to the information theoretic lower bound of log A =n for the description length of strings of length n. Using nondeterminism alone, we can achieve the information theoretic lower bound up to an additive term of O(( log A =n + log n) log n); using both nondeterminism and randomness, we can make do with an excess term of O(log 3 n). With randomness alone, we show a lower bound of n − log A =n − O(log n) on the description length of strings in A of length n, and a lower bound of 2 · log A =n − O(1) on the length of any program that distinguishes a given string of length n in A from any other string. The latter lower bound is tight up to an additive term of O(log n). The key ingredient for our upper bounds is the relativizable hardness versus randomness tradeoffs based on the Nisan-Wigderson pseudorandom generator construction.

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Simple Extractors via Constructions of Cryptographic Pseudo-random Generators

Zimand

2005

Automata, Languages and Programming

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Abstract. Trevisan has shown that constructions of pseudo-random generators from hard functions (the Nisan-Wigderson approach) also produce extractors. We show that constructions of pseudo-random generators from one-way permutations (the Blum-Micali-Yao approach) can be used for building extractors as well. Using this new technique we build extractors that do not use designs or polynomial-based error-correcting codes and that are very simple and efficient. For example, one extractor produces each output bit separately in O(log 2 n) time. These extractors work for weak sources with min entropy λn, for arbitrary constant λ > 0, have seed length O(log 2 n), and their output length is ≈ n λ/3 .

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Simple extractors for all min-entropies and a new pseudo-random generator

Cited by 78 publications

References 34 publications

Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

Language compression and pseudorandom generators

Simple Extractors via Constructions of Cryptographic Pseudo-random Generators

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