## Hardness amplification proofs require majority (2008)

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Venue: | In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC |

Citations: | 10 - 4 self |

### BibTeX

@INPROCEEDINGS{Shaltiel08hardnessamplification,

author = {Ronen Shaltiel and Emanuele Viola},

title = {Hardness amplification proofs require majority},

booktitle = {In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC},

year = {2008}

}

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### Abstract

Hardness amplification is the fundamental task of converting a δ-hard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)-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, ɛ, δ are small (and δ = 2 −k captures the case where f is worst-case hard). Achieving ɛ = 1/n ω(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 + ɛ fraction of the inputs there exists an oracle circuit D ∈ D such that D h agrees with f on a 1 − δ fraction of the inputs. We focus on the case where every D ∈ 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/ɛ bits. In particular, these circuits have large depth when ɛ ≤ 1/poly log n. 2. The circuits in D must make Ω � log(1/δ)/ɛ 2 � oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors.