On the complexity of hardness amplification

Abstract

For delta is an element of (0, 1) and k, n is an element of N, we study the task of transforming a hard function f : {0, 1}(n) -> {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(k1/d)) or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth) for any delta is an element of (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.