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.

Citations