Uniform hardness vs. randomness tradeoffs for Arthur-Merlin games

Dan Gutfreund; Ronen Shaltiel; Amnon Ta-Shma

Uniform hardness vs. randomness tradeoffs for Arthur-Merlin games

Dan Gutfreund^*, Ronen Shaltiel, Amnon Ta-Shma

^*Corresponding author for this work

School of Computer Science

Research output: Contribution to journal › Conference article › peer-review

6 Scopus citations

Abstract

Impagliazzo and Wigderson proved a uniform hardness vs. randomness "gap result" 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 sub-exponential time verifier, or else Arthur-Merlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm in the uniform average-case setting (i.e., it is infeasible to come up with inputs on which the algorithm fails). For the class AM ⊂ coAM we can remove the average-case clause and show under the same assumption that AM ⊂ coAM = NP ⊂ coNP. A new ingredient in our proof is identifying a novel resiliency property of hardness vs. randomness trade-offs. We observe that the Miltersen-Vinodchandran generator has this property.

Original language	English
Pages (from-to)	33-47
Number of pages	15
Journal	Proceedings of the Annual IEEE Conference on Computational Complexity
State	Published - 2003
Event	18th Annual IEEE Conference on Computational Complexity - Aarhus, Denmark Duration: 7 Jul 2003 → 10 Jul 2003

Cite this

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abstract = "Impagliazzo and Wigderson proved a uniform hardness vs. randomness {"}gap result{"} 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 sub-exponential time verifier, or else Arthur-Merlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm in the uniform average-case setting (i.e., it is infeasible to come up with inputs on which the algorithm fails). For the class AM ⊂ coAM we can remove the average-case clause and show under the same assumption that AM ⊂ coAM = NP ⊂ coNP. A new ingredient in our proof is identifying a novel resiliency property of hardness vs. randomness trade-offs. We observe that the Miltersen-Vinodchandran generator has this property.",

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AB - Impagliazzo and Wigderson proved a uniform hardness vs. randomness "gap result" 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 sub-exponential time verifier, or else Arthur-Merlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm in the uniform average-case setting (i.e., it is infeasible to come up with inputs on which the algorithm fails). For the class AM ⊂ coAM we can remove the average-case clause and show under the same assumption that AM ⊂ coAM = NP ⊂ coNP. A new ingredient in our proof is identifying a novel resiliency property of hardness vs. randomness trade-offs. We observe that the Miltersen-Vinodchandran generator has this property.

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Uniform hardness vs. randomness tradeoffs for Arthur-Merlin games

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