On the Complexity of Hardness Amplification

For deltaisin(0,1) and k , n isinN, we study the task of transforming a hard function f : {0,1} n rarr {0,1}, with which any small circuit disagrees on (1-delta)/2 fraction of the input, into a harder function f ', 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 deltaisin(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 ', 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.