Santha-Vazirani sources, deterministic condensers and very strong extractors

TL;DR

This paper investigates Santha-Vazirani sources, constructs deterministic condensers for them, and introduces the concept of very strong extractors, revealing their close relationship and advancing understanding of high-min-entropy source extraction.

Contribution

It constructs deterministic condensers for strong SV-sources and introduces very strong extractors, strengthening the connection between condensers and source unpredictability.

Findings

Deterministic condensers for strong SV-sources with high min-entropy rate.

Very strong extractors that maintain unpredictability even with knowledge of previous outputs.

Enhanced understanding of the relationship between condensers, extractors, and SV-sources.

Abstract

The notion of semi-random sources, also known as Santha-Vazirani (SV) sources, stands for a sequence of n bits, where the dependence of the i'th bit on the previous i-1 bits is limited for every $i\in[n]$. If the dependence of the i'th bit on the remaining n-1 bits is limited, then this is a strong SV-source. Even the strong SV-sources are known not to admit (universal) deterministic extractors, but they have seeded extractors, as their min-entropy is $\Omega(n)$. It is intuitively obvious that strong SV-sources are more than just high-min-entropy sources, and this work explores the intuition. Deterministic condensers are known not to exist for general high-min-entropy sources, and we construct for any constants $\epsilon, \delta \in (0,1)$ a deterministic condenser that maps n bits coming from a strong SV-source with bias at most $\delta$ to $\Omega(n)$ bits of min-entropy rate at least $1-\epsilon$. In conclusion we observe that deterministic condensers are closely related to very strong extractors - a proposed strengthening of the notion of strong (seeded) extractors: in particular, our constructions can be viewed as very strong extractors for the family of strong Santha-Vazirani distributions. The notion of very strong extractors requires that the output remains unpredictable even to someone who knows not only the seed value (as in the case of strong extractors), but also the extractor's outputs corresponding to the same input value with each of the preceding seed values (say, under the lexicographic ordering). Very strong extractors closely resemble the original notion of SV-sources, except that the bits must satisfy the unpredictability requirement only on average.