Is it possible to improve Yao's XOR lemma using reductions that exploit the efficiency of their oracle?

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Abstract

Yao's XOR lemma states that for every function f: {0, 1}k → {0, 1}, if f has hardness 2/3 for P/poly (meaning that for every circuit C in P/poly, Pr[C(X) = f(X)] ≤ 2/3 on a uniform input X), then the task of computing f(X1) ⊕... ⊕ f(Xt) for sufficiently large t has hardness 12 + ε for P/poly. Known proofs of this lemma cannot achieve ε = kω1(1), and even for ε = k1, we do not know how to replace P/poly by AC0[parity] (the class of constant depth circuits with the gates {and,or,not,parity} of unbounded fan-in). Recently, Grinberg, Shaltiel and Viola (FOCS 2018) (building on a sequence of earlier works) showed that these limitations cannot be circumvented by black-box reductions. Namely, by reductions Red(·) that given oracle access to a function D that violates the conclusion of Yao's XOR lemma, implement a circuit that violates the assumption of Yao's XOR lemma. There are a few known reductions in the related literature on worst-case to average case reductions that are non-black box. Specifically, the reductions of Gutfreund, Shaltiel and Ta Shma (Computational Complexity 2007) and Hirahara (FOCS 2018)) are “class reductions” that are only guaranteed to succeed when given oracle access to an oracle D from some efficient class of algorithms. These works seem to circumvent some black-box impossibility results. In this paper we extend the previous limitations of Grinberg, Shaltiel and Viola to class reductions, giving evidence that class reductions cannot yield the desired improvements in Yao's XOR lemma. To the best of our knowledge, this is the first limitation on reductions for hardness amplification that applies to class reductions. Our technique imitates the previous lower bounds for black-box reductions, replacing the inefficient oracle used in that proof, with an efficient one that is based on limited independence, and developing tools to deal with the technical difficulties that arise following this replacement.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020
EditorsJaroslaw Byrka, Raghu Meka
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771641
DOIs
StatePublished - 1 Aug 2020
Event23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020 - Virtual, Online, United States
Duration: 17 Aug 202019 Aug 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume176
ISSN (Print)1868-8969

Conference

Conference23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020
Country/TerritoryUnited States
CityVirtual, Online
Period17/08/2019/08/20

Bibliographical note

Funding Information:
This research was supported by ISF grant 1628/17.

Publisher Copyright:
© 2020 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

  • Black-box reductions
  • Hardness amplification
  • Yao's XOR lemma

ASJC Scopus subject areas

  • Software

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