Hitting sets derandomize BPP

Andreev, Alexander E.; Clementi, Andrea E. F.; Rolim, José D. P.

doi:10.1007/3-540-61440-0_142

Cited by 24 publications

(18 citation statements)

References 8 publications

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“…Theorem (informal) 1. Either E = DTIME (2 O( ) ) is computable by ArthurMerlin protocols with time s( ) or for any AM language L there is a nondeterministic machine M that runs in time exponential in and solves L correctly on feasibly generated inputs of length n = s( ) Θ(1/(log −log log s( ))…”

Section: Our Resultmentioning

confidence: 99%

“…It is standard that given an HSG against deterministic (resp., co-nondeterministic) circuits of size poly(m) one can derandomize RP (resp., AM) in time poly(T ) by simulating the algorithm (resp., protocol) on all strings output by the HSG, and accepting if at least one of the runs accepts. It is also known that HSGs against deterministic circuits suffice to derandomize two-sided error (BPP) [1,2].…”

Section: Pseudorandom Generators and Hitting Set Generatorsmentioning

confidence: 99%

“…The running time also suffers as each recursive step multiplies the running time of the protocol by poly(m). When taking these two factors into consideration and transforming to a two round AM protocol, we get that this protocol has running time m 1) . This accounts for the slight nonoptimality of our main gap theorems.…”

Section: Losses Suffered In the Recursionmentioning

confidence: 99%

“…Recall that in the definition of M L we chose to be the smallest integer such that where we are using the fact that (log v( ) − log log m) = O(log v( ) − log log s( )), which follows because by the definition of m (log v( ) − log log m) = (log v( ) − log log s( )) + 2 log (log v( ) − log log s( )) + O (1) ≤ (1 + o(1))(log v( ) − log log s( )) + O(1).…”

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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: Our Resultmentioning

confidence: 99%

Section: Pseudorandom Generators and Hitting Set Generatorsmentioning

confidence: 99%

Section: Losses Suffered In the Recursionmentioning

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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“…We then constructed all candidate generators in the tree, and showed how to combine them into a hitting set generator with optimal seed length. This in turn suffices for derandomizing two-sided error probabilistic algorithms using the results of [ACR96,ACRT99]. We also gave a more direct argument in which we conduct a "tournament" between all the candidate pseudo-random generators.…”

Section: History Of This Papermentioning

confidence: 99%

Reducing The Seed Length In The Nisan-Wigderson Generator*

2006

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The Nisan-Wigderson pseudo-random generator [NW94] was constructed to derandomize probabilistic algorithms under the assumption that there exist explicit functions which are hard for small circuits. We give the first explicit construction of a pseudo-random generator with asymptotically optimal seed length even when given a function which is hard for relatively small circuits. Generators with optimal seed length were previously known only assuming hardness for exponential size circuits [IW97,STV01].We also give the first explicit construction of an extractor which uses asymptotically optimal seed length for random sources of arbitrary min-entropy. Our construction is the first to use the optimal seed length for sub-polynomial entropy levels. It builds on the fundamental connection between extractors and pseudo-random generators discovered by Trevisan [Tre01], combined with the construction above.The key is a new analysis of the NW-generator [NW94]. We show that it fails to be pseudo random only if a much harder function can be efficiently constructed from the given hard function. By repeatedly using this idea we get a new recursive generator, which may be viewed as a reduction from the general case of arbitrary hardness to the solved case of exponential hardness. * This paper is based on two conference papers [ISW99, ISW00] by the same authors.

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Derandomizing Arthur-Merlin Games under Uniform Assumptions

2000

Algorithms and Computation

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Hitting sets derandomize BPP

Cited by 24 publications

References 8 publications

Low-End Uniform Hardness versus Randomness Tradeoffs for AM

Low-End Uniform Hardness versus Randomness Tradeoffs for AM

Reducing The Seed Length In The Nisan-Wigderson Generator*

Derandomizing Arthur-Merlin Games under Uniform Assumptions

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