Measure of Non-pseudorandomness and Deterministic Extraction of Pseudorandomness

In this paper, we propose a quantification of distributions on a set of strings, in terms of how close to pseudorandom the distribution is. The quantification is an adaptation of the theory of dimension of sets of infinite sequences first introduced by Lutz [13]. We show that this definition is robust, by considering an alternate, equivalent quantification. It is known that pseudorandomness can be characterized in terms of predictors [20]. Adapting Hitchcock [10], we show that the log-loss function incurred by a predictor on a distribution is quantitatively equivalent to the notion of dimension we define. We show that every distribution on a set of strings of length n has a dimension s ∈ [0, 1], and for every s ∈ [0, 1] there is a distribution with dimension s. We study some natural properties of our notion of dimension. Further, we propose an application of our quantification to the following problem. If we know that the dimension of a distribution on the set of n-length strings is s ∈ [0, 1], can we deterministically extract out sn pseudorandom bits out of the distribution? We show that this is possible in a special case a notion analogous to the bit-fixing sources introduced by Chor et. al. [5], which we term a nonpseudorandom bit-fixing source. We adapt the techniques of Kamp and Zuckerman [12] and Gabizon, Raz and Shaltiel [7] to establish that in the case of a non-pseudorandom bit-fixing source, we can deterministically extract the pseudorandom part of the source. Further, we show that the existence of optimal nonpseudorandom generator is enough to show P = BPP. ∗Research supported by Research-I Foundation †[email protected] ‡[email protected] §[email protected] ¶[email protected]