Deterministic Randomness Extraction from Generalized and Distributed Santha-Vazirani Sources

TL;DR

This paper explores the possibility of deterministic randomness extraction from generalized and distributed Santha-Vazirani sources, providing conditions under which extraction is feasible, and extends classical results to adversarial and non-binary settings.

Contribution

It introduces necessary and sufficient conditions for deterministic extraction in non-binary and distributed SV sources, generalizing previous binary and i.i.d. results.

Findings

Deterministic extraction is sometimes possible for non-binary SV sources.

Conditions for extraction coincide in non-degenerate cases.

Reduces distributed extraction problem to non-distributed case using maximal correlation.

Abstract

A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for non-binary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in "non-degenerate" cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of G\'acs and K\"orner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.