On the Complexity of Hardness Amplification

For$\delta \in (0,1)$and$k,n\in \BBN $, we study the task of transforming a hard function$f: \{0,1\}^{n}\to \{0,1\} $, with which any small circuit disagrees on$(1-\delta )/2$fraction of the input, into a harder function$f^{\prime}$, with which any small circuit disagrees on$(1-\delta ^{k})/2$fraction of the input. First, we show that such hardness amplification, when carried out in some black-box way, must require a high complexity. In particular, it cannot be realized by a circuit of depth$d$and size$2^{o(k^{1/d})}$or by a nondeterministic circuit of size$o(k/\log k)$(and arbitrary depth) for any$\delta \in (0,1)$. This extends the result of Viola, which only works when$(1-\delta )/2$is small enough. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently nonuniform in the following sense. To guarantee the hardness of the resulting function$f^{\prime}$, even against uniform machines, one has to start with a function$f$, which is hard against nonuniform algorithms with$\Omega (k\log (1/\delta ))$bits of advice. This extends the result of Trevisan and Vadhan, which only addresses the case with$(1-\delta )/2=2^{-n}$. Finally, we derive similar lower bounds for any black-box construction of a pseudorandom generator (PRG) from a hard function. To prove our results, we link the task of hardness amplifications and PRG constructions, respectively, to some type of error-reduction codes, and then we establish lower bounds for such codes, which we hope could find interest in both coding theory and complexity theory.