Computational Randomness from Generalized Hardcore Sets

Chia-Jung Lee, Chi-Jen Lu, Shi-Chun Tsai

Abstract: The seminal hardcore lemma of Impagliazzo states that for any mildly-hard Boolean function f, there is a subset of input, called the hardcore set, on which the function is extremely hard, almost as hard as a random Boolean function. This implies that the output distribution of f given a random input looks like a distribution with some statistical randomness. Can we have something similar for hard functions with several output bits? Can we say that the output distribution of such a general function given a random input looks like a distribution containing several bits of randomness? If so, one can simply apply any statistical extractor to extract computational randomness from the output of f. However, the conventional wisdom tells us to apply extractors with some additional reconstruction property, instead of just any extractor. Does this mean that there is no analogous hardcore lemma for general functions?