On the Complexity of Hardness Amplification

We study the task of transforming a hard function f, with which any small circuit disagrees on (1 - )/2 fraction of the input, into a harder function f , with which any small circuit disagrees on (1 - k )/2 fraction of the input, for (0, 1) and k N. We show that this process can not be carried out in a black-box way by a circuit of depth d and size 2o(k1/d ) or by a nondeterministic circuit of size o(k/ log k) (and arbitrary depth). In particular, for k = 2(n) , such hardness amplification can not be done in ATIME(O(1), 2o(n) ). Therefore, hardness amplification in general requires a high complexity. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently non-uniform in the following sense. Given as an oracle any algorithm which agrees with f on (1 - k )/2 fraction of the input, we still need an additional advice of length (k log(1/)) in order to compute f correctly on (1-)/2...

• Type: Journal
• Year: 2008
• Where: TIT
• Authors: Chi-Jen Lu, Shi-Chun Tsai, Hsin-Lung Wu
• Keywords: Hard Functions, Input, Small Circuit, TIT 2008