TY - GEN

T1 - Low-end uniform hardness vs. randomness tradeoffs for AM

AU - Shaltiel, Ronen

AU - Umans, Christopher

PY - 2007

Y1 - 2007

N2 - In 1998, Impagliazzo and Wigderson [18] proved a hardness vs. randomness tradeoff for BPP in the uniform setting, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions by Trevisan and Vadhan [29] (in a slightly weaker setting). In 2003, Gutfreund, Shaltiel and Ta-Shma [11] proved a uniform hardness vs. randomness tradeoff for AM, but result only worked on the "high-end" of hardness assumptions. In this work, we uniform hardness vs. randomness tradeoffs for AM that are near-optimal for the full range of possible hardness assumptions. Following [11], we do this by constructing a hitting-set-generator (HSG) for AM with "resilient reconstruction." Our construction is a recursive variant of the Miltersen-Vinodchandran HSG [24], 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 [28, 26] for a sort of locallycomputable error-duction. As a consequence we obtain gap theorems for AM (and AM \ coAM) that state, roughly, that either AM (or AM ∩ coAM)rotocols 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 [11], 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.

AB - In 1998, Impagliazzo and Wigderson [18] proved a hardness vs. randomness tradeoff for BPP in the uniform setting, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions by Trevisan and Vadhan [29] (in a slightly weaker setting). In 2003, Gutfreund, Shaltiel and Ta-Shma [11] proved a uniform hardness vs. randomness tradeoff for AM, but result only worked on the "high-end" of hardness assumptions. In this work, we uniform hardness vs. randomness tradeoffs for AM that are near-optimal for the full range of possible hardness assumptions. Following [11], we do this by constructing a hitting-set-generator (HSG) for AM with "resilient reconstruction." Our construction is a recursive variant of the Miltersen-Vinodchandran HSG [24], 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 [28, 26] for a sort of locallycomputable error-duction. As a consequence we obtain gap theorems for AM (and AM \ coAM) that state, roughly, that either AM (or AM ∩ coAM)rotocols 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 [11], 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.

KW - Arthur-Merlin games

KW - Derandomization

KW - Hardness vs. randomness tradeoff

KW - Hitting-set generator

UR - http://www.scopus.com/inward/record.url?scp=35448955268&partnerID=8YFLogxK

U2 - 10.1145/1250790.1250854

DO - 10.1145/1250790.1250854

M3 - Conference contribution

AN - SCOPUS:35448955268

SN - 1595936319

SN - 9781595936318

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 430

EP - 439

BT - STOC'07

T2 - STOC'07: 39th Annual ACM Symposium on Theory of Computing

Y2 - 11 June 2007 through 13 June 2007

ER -