## Abstract

Hardness amplification is the fundamental task of converting a δ-hard function f : {0,1}^{n} → {0,1} into a (1/2 - ε)-hard function Amp(f), where f is γ-hard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ε, δ are small (and δ = 2^{-k} captures the case where f is worst-case hard). Achieving ε = l/n^{w(1)} is a prerequisite for cryptography and most pseudorandom-generator constructions. In this paper we study the complexity of black-box proofs of hardness amplification. A class of circuits D proves a hardness amplification result if for any function h that agrees with Amp(f) on a 1/2 + ε fraction of the inputs there exists an oracle circuit D ε D such that D^{h} agrees with f on a 1 - δ fraction of the inputs. We focus on the case where every D ε D makes non-adaptive queries to h. This setting captures most hardness amplification techniques. We prove two main results: 1. The circuits in D "can be used" to compute the majority function on 1/ε bits. In particular, these circuits have large depth when ε ≤ 1 /poly log n. 2. The circuits in D must make ω. (log(l/δ)/ε^{2}) oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors. Our results explain why hardness amplification techniques have failed to transform known lower bounds against constant-depth circuit classes into strong average-case lower bounds. When coupled with the celebrated "Natural Proofs" result by Razborov and Rudich (J. CSS '97) and the pseudorandom functions by Naor and Reingold (J. ACM '04), our results show that standard techniques for hardness amplification can only be applied to those circuit classes for which standard techniques cannot prove circuit lower bounds. Our results reveal a contrast between Yao's XOR. Lemma (Amp(f) := f(x_{1}) o ... o f(x_{t}) ε {0, 1}) and the Direct-Product Lemma (Amp(f) := f(x_{i}) o ... o f(x_{t}) ε {0,1}^{t}; here Amp(f) is non-Boolean). Our results (1) and (2) apply to Yao's XOR lemma, whereas known proofs of the direct-product lemma violate both (1) and (2). One of our contributions is a, new technique to handle "non-uniform" reductions, i.e. the case when D contains many circuits.

Original language | English |
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Title of host publication | STOC'08 |

Subtitle of host publication | Proceedings of the 2008 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery (ACM) |

Pages | 589-598 |

Number of pages | 10 |

ISBN (Print) | 9781605580470 |

DOIs | |

State | Published - 2008 |

Event | 40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada Duration: 17 May 2008 → 20 May 2008 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 40th Annual ACM Symposium on Theory of Computing, STOC 2008 |
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Country/Territory | Canada |

City | Victoria, BC |

Period | 17/05/08 → 20/05/08 |

### Bibliographical note

Funding Information:This study was supported by National Natural Science Foundation of China (81572536, 81672850), Science and Technology Commission of Shanghai Municipality (14140901700, 16411969800), the Joint Research Foundation for Innovative Medical Technology of Shanghai Shenkang Hospital Development Center (SHDC12015125), Shanghai Municipal Education Commission (15ZZ058), Shanghai Municipal Commission of Health and Family Planning (201640247), Shanghai Municipal Education Commission- Gaofeng Clinical Medicine Grant Support (20152215), Key Disciplines Group Construction Project of Pudong Health Bureau of Shanghai (PWZxq2014-05), Innovation Fund for Translational Research of Shanghai Jiao Tong University School of Medicine (15ZH4002), and Incubating Program for clinical Research and Innovation of Renji Hospital Shanghai Jiao Tong University School of Medicine (PYZY 16-008, PYXJS16-015).

## Keywords

- Average-case complexity
- Black-box
- Constant-depth circuits
- Hardness amplification
- Majority
- Natural proofs

## ASJC Scopus subject areas

- Software