On the complexity of hardness amplification

We study the task of transforming a hard function f, with which any small circuit disagrees on (1 - δ)/2 fraction of the input, into a harder function f´, with which any small circuit disagrees on (1 - δ^k)/2 fraction of the input, for δ ∈ (0,1) and k ∈ N. We show that this process cannot be carried out in a black-box way by a circuit of depth d and size 2^o(k2d)/ or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth). In particular, for k = 2^Ω(n), such hardness amplification cannot be done in ATIME(O(1), 2^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 - δ^k)/2 fraction of the input, we still need an additional advice of length Ω(k log(1/δ)) in order to compute f correctly on (1 - δ)/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.