Computational Analogues of Entropy

Min-entropy is a statistical measure of the amount of randomness that a particular distribution contains. In this paper we investigate the notion of computational min-entropy which is the computational analog of statistical min-entropy. We consider three possible definitions for this notion, and show equivalence and separation results for these definitions in various computational models. We also study whether or not certain properties of statistical min-entropy have a computational analog. In particular, we consider the following questions: 1. Let X be a distribution with high computational min-entropy. Does one get a pseudorandom distribution when applying a “randomness extractor” on X? 2. Let X and Y be (possibly dependent) random variables. Is the computational min-entropy of (X,Y ) at least as large as the computational min-entropy of X? 3. Let X be a distribution over {0, 1} that is “weakly unpredictable” in the sense that it is hard to predict a constant fraction of the coordinates of X with a constant bias. Does X have computational min-entropy Ω(n)? We show that the answers to these questions depend on the computational model considered. In some natural models the answer is false and in others the answer is true. Our positive results for the third question exhibit models in which the “hybrid argument bottleneck” in “moving from a distinguisher to a predictor” can be avoided.