Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

Dan Gutfreund; Ronen Shaltiel; Amnon Ta-Shma

doi:10.1007/s00037-003-0178-7

Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

Dan Gutfreund
, Ronen Shaltiel
, Amnon Ta-Shma

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	85-130
Number of pages	46
Journal	Computational Complexity
Volume	12
Issue number	3-4
DOIs	https://doi.org/10.1007/s00037-003-0178-7
State	Published - 2003
Externally published	Yes

Bibliographical note

Funding Information:
The first author is supported in part by the Leibniz Center, the Israel Foundation of Science, a US-Israel Binational research grant, and an EU Information Technologies grant (IST-FP5). The second author is supported by the Koshland Scholarship.

Keywords

Arthur-Merlin games
Derandomization

ASJC Scopus subject areas

Theoretical Computer Science
General Mathematics
Computational Theory and Mathematics
Computational Mathematics

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10.1007/s00037-003-0178-7

Cite this

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title = "Uniform hardness versus randomness tradeoffs for Arthur-Merlin games",

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.",

keywords = "Arthur-Merlin games, Derandomization",

author = "Dan Gutfreund and Ronen Shaltiel and Amnon Ta-Shma",

note = "Funding Information: The first author is supported in part by the Leibniz Center, the Israel Foundation of Science, a US-Israel Binational research grant, and an EU Information Technologies grant (IST-FP5). The second author is supported by the Koshland Scholarship.",

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doi = "10.1007/s00037-003-0178-7",

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AU - Shaltiel, Ronen

AU - Ta-Shma, Amnon

N1 - Funding Information: The first author is supported in part by the Leibniz Center, the Israel Foundation of Science, a US-Israel Binational research grant, and an EU Information Technologies grant (IST-FP5). The second author is supported by the Koshland Scholarship.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - Arthur-Merlin games

KW - Derandomization

UR - https://www.scopus.com/pages/publications/4744337602

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JO - Computational Complexity

JF - Computational Complexity

IS - 3-4

ER -

Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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