## Abstract

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 language | American English |
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Pages (from-to) | 43-83 |

Number of pages | 41 |

Journal | Computational Complexity |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2014 |

## Keywords

- Hardness amplification
- black-box reductions

## All Science Journal Classification (ASJC) codes

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