Lower Bounds on the Query Complexity of Non-uniform and Adaptive Reductions Showing Hardness Amplification

Sergei Artemenko, Ronen Shaltiel

Research output: Contribution to journalArticlepeer-review


Hardness amplification results show that for every Boolean function f, there exists a Boolean function Amp(f) such that if every size s circuit computes f correctly on at most a 1 - δ fraction of inputs, then every size s′ circuit computes Amp(f) correctly on at most a 1/2 + ∈ fraction of inputs. All hardness amplification results in the literature suffer from "size loss" meaning that s′ ≤ ∈ · s. We show that proofs using "non-uniform reductions" must suffer from such size loss. A reduction is an oracle circuit R(·) which given oracle access to any function D that computes Amp(f) correctly on a 1/2 + ∈ fraction of inputs, computes f correctly on a 1 - δ fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string that may depend on both f and D. The well-known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for ∈ < 1/4. We show that every non-uniform reduction must make at least Ω(1/∈) queries to its oracle, which implies size loss. Our result is the first lower bound that applies to non-uniform reductions that are adaptive, whereas previous bounds by Shaltiel & Viola (SICOMP 2010) applied only to non-adaptive reductions. We also prove similar bounds for a stronger notion of "function-specific" reductions in which the reduction is only required to work for a specific function f.

Original languageEnglish
Pages (from-to)43-83
Number of pages41
JournalComputational Complexity
Issue number1
StatePublished - Mar 2014

Bibliographical note

Funding Information:
A preliminary version of this paper appeared in Proceedings of the 15th International Workshop on Randomization and Computation, p. 377-388, 2011. This research was supported by BSF grant 2010120, ISF grants 686/07 and 864/11, and ERC starting grant 279559. The second author is grateful to Oded Goldreich, Avi Wigderson and Emanuele Viola for many interesting discussions on hardness amplification. We also thank Oded Goldreich, Danny Gutfreund, Iftach Haitner and anonymous referees for helpful comments and suggestions.


  • Hardness amplification
  • black-box reductions

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Mathematics
  • Computational Theory and Mathematics
  • Computational Mathematics


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  • Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification

    Artemenko, S. & Shaltiel, R., 2011, Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. p. 377-388 12 p. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); vol. 6845 LNCS).

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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