Indistinguishability and Unpredictability Hardcore Lemmas: New Proofs with Applications to Pseudoentropy

Hardcore lemmas are results in complexity theory which state that average-case hardness must have a very hard “kernel”, that is a subset of instances where the problem is extremely hard. Such results find important applications in hardness amplification. In this paper we revisit two classical results: (a) The hardcore lemma for unpredictability, proved first by Impagliazzo. It states that if a boolean function f is “moderately” hard to predict on average, then there must be a set of noticeable size on which f is “extremely” hard to predict. (b) The hardcore lemma for indistnguishability, proved by Maurer and Tesaro, states that for two random variables X and Y which are -computationally close, there exist events A and B of probability 1− such that the distributions of X|A and Y |B are “almost” identical. We provide alternative proofs and some generalizations of these result in the nonuniform setting. As an interesting application, we show a strengthening of the transformation between two most popular pseudoentropy variants: HILL and Metric Entropy, and apply it to show how to extract pseudorandomness from a sequence of metric-entropy sources of poor quality. Comparing to the best known techniques we significantly improve security parameters.