Hardness Amplification Proofs Require Majority

Hardness amplification is the fundamental task of converting a -hard function f : {0, 1}n {0, 1} into a (1/2 - )-hard function Amp(f), where f is -hard if small circuits fail to compute f on at least a fraction of the inputs. Typically, , are small (and = 2-k captures the case where f is worst-case hard). Achieving = 1/n(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 + fraction of the inputs there exists an oracle circuit D D such that Dh agrees with f on a 1- 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:

• Type: Journal
• Year: 2010
• Where: SIAMCOMP
• Authors: Ronen Shaltiel, Emanuele Viola
• Keywords: -hard Function, Hardness Amplification, Hardness Amplification Techniques, SIAMCOMP 2010