Uniform direct product theorems: Simplified, optimized, and derandomized

Russell Impagliazzo*, Ragesh Jaiswal, Valentine Kabanets, Avi Wigderson

*Corresponding author for this work

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

32 Scopus citations

Abstract

The classical Direct-Product Theorem for circuits says that if a Boolean function f : {0, 1}n → {0. 1} is somewhat hard to compute on average by small circuits, then the corresponding fc-wise direct product function fk (x1,..., xk) = (f(x 1),..., f(xk)) (where each xi ε {0, 1}n) is significantly harder to compute on average by slightly smaller circuits. We prove a, fully uniform version of the Direct-Product Theorem with information-theoretically optimal parameters, up to constant factors. Namely, we show that for given k and ε, there is an efficient randomized algorithm A with the following property. Given a circuit G that computes fk on at least e fraction of inputs, the algorithm A outputs with probability at least 3/4 a, list of O(1/ε) circuits such that at least one of the circuits on the list computes / on more than 1 - δ fraction of inputs, for δ = O((log 1/ε)/k). Moreover, each output circuit is an ACo circuit (of size poly(n, k, log 1/δ,1/ε)), with oracle access to the circuit C. Using the Goldreich-Levin decoding algorithm [5], we also get a fully uniform version of Yao's XOR, Lemma [18] with optimal parameters, up to constant factors. Our results simplify and improve those in [10]. Our main result may be viewed as an efficient approximate, local, list-decoding algorithm for direct-product codes (encoding a function by its values on all fc-tuples) with optimal parameters. We generalize it to a family of "derandomized" direct-product codes, which we call intersection codes, where the encoding provides values of the function only on a subfamily of fe-tuples. The quality of the decoding algorithm is then determined by sampling properties of the sets in this family and their intersections. As a direct consequence of this generalization we obtain the first derandomized direct product result in the uniform setting, allowing hardness amplification with only constant (as opposed to a factor of fc) increase in the input length. Finally, this general setting naturally allows the decoding of concatenated codes, which further yields nearly optimal derandomized amplification.

Original languageEnglish
Title of host publicationSTOC'08
Subtitle of host publicationProceedings of the 2008 ACM Symposium on Theory of Computing
Pages579-588
Number of pages10
StatePublished - 2008
Externally publishedYes
Event40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada
Duration: 17 May 200820 May 2008

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference40th Annual ACM Symposium on Theory of Computing, STOC 2008
Country/TerritoryCanada
CityVictoria, BC
Period17/05/0820/05/08

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