## Abstract

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},..., x_{k}) = f(x_{1})⊕...⊕f(x _{k}) on independently chosen x_{1},..., x_{k}. 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 language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Computational Complexity |

Volume | 12 |

Issue number | 1-2 |

DOIs | |

State | Published - 2003 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

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