Towards proving strong direct product theorems

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A fundamental question of complexity theory is the direct product question. A famous example is Yao's XOR-lemma, in which one assumes that some function f is hard on average for small circuits (mean-ing that every circuit of some fixed size s which attempts to compute f is wrong on a non-negligible fraction of the inputs) and concludes that every circuit of size s′ only has a small advantage over guessing randomly when computing f⊕k (x 1,..., xk) = f(x1)⊕...⊕f(x k) on independently chosen x1,..., xk. All known proofs of this lemma have the property that s′ < s. In words, the circuit which attempts to compute f⊕k is smaller than the circuit which attempts to compute f on a single input! This paper addresses the issue of proving strong direct product assertions, that is, ones in which s′ ≈ ks and is in particular larger than s. We study the question of proving strong direct product question for decision trees and communication protocols.

Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalComputational Complexity
Issue number1-2
StatePublished - 2003
Externally publishedYes


  • Average case complexity
  • Hardness amplification
  • Product theorems
  • XOR-lemma

ASJC Scopus subject areas

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


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