Reducing The Seed Length In The Nisan-Wigderson Generator*

Impagliazzo, Russell; Shaltiel, Ronen; Wigderson, Avi

doi:10.1007/s00493-006-0036-8

Cited by 17 publications

(10 citation statements)

References 38 publications

(57 reference statements)

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“…A discussion regarding the best possible parameters that can be expected in hardness versus randomness tradeoffs appears in [18]. Our results are suboptimal in the sense that one could hope to get n = s( ) Ω(1) whereas we get only n = s( )…”

Section: Theorem (Informal) 2 Either E = Dtime(2 O( ) ) Is Computablmentioning

confidence: 85%

“…Although it has long been observed that there is some similarity between aspects of PCP constructions and aspects of PRG and HSG constructions, this seems to be the first time primitives like low-degree testing have proven useful in such constructions. Second, we run both the construction of [27] and the associated reduction recursively, in a manner reminiscent of [18,35] (although the low-level details are different). Finally, we introduce a new primitive called local extractors for Reed-Muller codes, which are extractors that are computable in sublinear time when run on inputs that are guaranteed to be ReedMuller codewords.…”

Section: Our Techniquesmentioning

confidence: 99%

“…However, the "low end" one assumes hardness against smaller circuits of size s( ) = poly( ) and can conclude "weak derandomization," i.e., simulations of BPP (resp., AM) that run in subexponential deterministic (resp., nondeterministic subexponential) time [6,30]. Today, after a long line of research [28,6,16,19,3,23,27,18,30,34,29,35] we have optimal hardness versus randomness tradeoffs for both BPP and AM that achieve "optimal parameters" in the nonuniform setting (see the discussion of nonuniform versus uniform below).…”

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confidence: 99%

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Low-End Uniform Hardness versus Randomness Tradeoffs for AM

Shaltiel¹,

Umans²

2009

SIAM J. Comput.

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Abstract. Impagliazzo and Wigderson [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, 1998, pp. 734-743] proved a hardness versus randomness tradeoff for BPP in the uniform setting, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions (in slightly weaker settings). Gutfreund, Shaltiel, and Ta-Shma [Comput. Complexity, 12 (2003), pp. 85-130] proved a uniform hardness versus randomness tradeoff for AM, but that result worked only on the "high end" of possible hardness assumptions. In this work, we give uniform hardness versus randomness tradeoffs for AM that are near-optimal for the full range of possible hardness assumptions. Following Gutfreund, Shaltiel, and Ta-Shma, we do this by constructing a hitting-set-generator (HSG) for AM with "resilient reconstruction." Our construction is a recursive variant of the MiltersenVinodchandran HSG [Comput. Complexity, 14 (2005), pp. 256-279], the only known HSG construction with this required property. The main new idea is to have the reconstruction procedure operate implicitly and locally on superpolynomially large objects, using tools from PCPs (low-degree testing, self-correction) together with a novel use of extractors that are built from Reed-Muller codes for a sort of locally computable error-reduction. As a consequence we obtain gap theorems for AM (and AM ∩ coAM) that state, roughly, that either AM (or AM ∩ coAM) protocols running in time t(n) can simulate all of EXP ("Arthur-Merlin games are powerful") or else all of AM (or AM ∩ coAM) can be simulated in nondeterministic time s(n) ("Arthur-Merlin games can be derandomized") for a near-optimal relationship between t(n) and s(n). As in Gutfreund, Shatiel, and Ta-Shma, the case of AM ∩ coAM yields a particularly clean theorem that is of special interest due to the wide array of cryptographic and other problems that lie in this class.

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Section: Theorem (Informal) 2 Either E = Dtime(2 O( ) ) Is Computablmentioning

confidence: 85%

Section: Our Techniquesmentioning

confidence: 99%

mentioning

confidence: 99%

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Low-End Uniform Hardness versus Randomness Tradeoffs for AM

Shaltiel¹,

Umans²

2009

SIAM J. Comput.

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“…The main thread of this line of research dates back to the work of Shamir [4], Yao [5] and Blum & Micali [6], and involves showing that, given a suitably hard function f , one can construct pseudorandom generators and hitting-set generators. Much of the progress on this front over the years has involved showing how to weaken the hardness assumption on f and still obtain useful derandomizations [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In rare instances, it has been possible to obtain unconditional derandomizations using this framework; Nisan [23], Nisan & Wigderson [24] and Viola [25] showed that uniform families of probabilistic AC 0 circuits can be simulated by uniform deterministic AC 0 circuits of size n log O (1) n .…”

Section: Introductionmentioning

confidence: 99%

Uniform derandomization from pathetic lower bounds

Allender

Arvind

Santhanam

et al. 2012

Phil. Trans. R. Soc. A.

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The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where 'pathetic' lower bounds of the form n 1+e would suffice to derandomize interesting classes of probabilistic algorithms. We show the following:-If the word problem over S 5 requires constant-depth threshold circuits of size n 1+e for some e > 0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size). -If there are no constant-depth arithmetic circuits of size n 1+e for the problem of multiplying a sequence of n 3 × 3 matrices, then, for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

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“…Surprisingly, it was shown in [1] (and refined in [2,4,6]) that HSGs suffice to derandomize BPP, even though they are only "intended" for onesided error. Optimal HSGs were first constructed in [18], while optimal PRGs were first constructed in [26]; here "optimal" means that, up to a polynomial, the constructions cannot be improved without implying stronger hardness assumptions than were used to construct them in the first place (see [13] for a more detailed justification of the term "optimal" in this context).…”

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confidence: 99%

Reconstructive Dispersers and Hitting Set Generators

Umans

2008

Algorithmica

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We give a generic construction of an optimal hitting set generator (HSG) from any good "reconstructive" disperser. Past constructions of optimal HSGs have been based on such disperser constructions, but have had to modify the construction in a complicated way to meet the stringent efficiency requirements of HSGs. The construction in this paper uses existing disperser constructions with the "easiest" parameter setting in a black-box fashion to give new constructions of optimal HSGs without any additional complications.Our results show that a straightforward composition of the Nisan-Wigderson pseudorandom generator that is similar to the composition in works by Impagliazzo, Shaltiel and Wigderson in fact yields optimal HSGs (in contrast to the "near-optimal" HSGs constructed in those works). Our results also give optimal HSGs that do not use any form of hardness amplification or implicit list-decoding-like Trevisan's extractor, the only ingredients are combinatorial designs and any good list-decodable error-correcting code.

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Reducing The Seed Length In The Nisan-Wigderson Generator*

Cited by 17 publications

References 38 publications

Low-End Uniform Hardness versus Randomness Tradeoffs for AM

Low-End Uniform Hardness versus Randomness Tradeoffs for AM

Uniform derandomization from pathetic lower bounds

Reconstructive Dispersers and Hitting Set Generators

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