Note: Improved hardness amplification in NP
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We study the task of transforming a hard function f, with which any small circuit disagrees on (1-驴)/2 fraction of the input, into a harder function f驴, with which any small circuit disagrees on (1-驴^k)/2 fraction of the input, for 驴 驴 (0, 1) and k 驴 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(k^1/d) or by a nondeterministic circuit of size o(k/ log k) (and arbitrary depth). In particular, for k = 2^驴(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驴 on (1 - 驴^k)/2 fraction of the input, we still need an additional advice of length 驴(k log(1/驴)) in order to compute f correctly on (1 - 驴)/2 fraction of the input. Therefore, to guarantee the hardness, even against uniform machines, of the function f驴, 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.