On the Complexity of Hardness Amplification

Abstract

We study the task of transforming a hard function f, with which any small circuit disagrees on (1 - /spl delta/)/2 fraction of the input, into a harder function f', with which any small circuit disagrees on (1 - /spl delta//sup k/)/2 fraction of the input, for /spl delta/ /spl isin/ (0,1) and k /spl isin/ /spl Nopf/. We show that this process cannot be carried out in a black-box way by a circuit of depth d and size 2/sup o(k2/d)/ or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth). In particular, for k = 2/sup /spl Omega/(n)/, such hardness amplification cannot be done in ATIME(O(1), 2/sup o(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 - /spl delta//sup k/)/2 fraction of the input, we still need an additional advice of length /spl Omega/(k log(1//spl delta/)) in order to compute f correctly on (1 - /spl delta/)/2 fraction of the input. Therefore, to guarantee the hardness, even against uniform machines, of the function f', one has to start with a function f which is hard against non-uniform circuits. Finally, we derive similar lower bounds for any black-box construction of pseudorandom generators from hard functions.