On the Complexity of Hardness Amplification

Chi-Jen Lu, Shi-Chun Tsai, Hsin-Lung Wu
2008 IEEE Transactions on Information Theory  
For 2 (0; 1) and k; n 2 , we study the task of transforming a hard function f : f0; 1g n ! f0; 1g, with which any small circuit disagrees on (1 0 )=2 fraction of the input, into a harder function f 0 , with which any small circuit disagrees on (1 0 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 ) or by a
more » ... c circuit of size o(k= log k) (and arbitrary depth) for any 2 (0; 1). This extends the result of Viola, which only works when (1 0 )=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 0 , even against uniform machines, one has to start with a function f , which is hard against nonuniform algorithms with (k log(1=)) bits of advice. This extends the result of Trevisan and Vadhan, which only addresses the case with (1 0 )=2 = 2 0n . 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.
doi:10.1109/tit.2008.928988 fatcat:7niutgi45vdwhcuhg66cr2lgkq