Santha-Vazirani sources, deterministic condensers and very strong extractors

0531290 - MÚ 2021 RIV US eng J - Journal Article
Gavinsky, Dmitry - Pudlák, Pavel
Santha-Vazirani sources, deterministic condensers and very strong extractors.
Theory of Computing Systems. Roč. 64, č. 6 (2020), s. 1140-1154. ISSN 1432-4350. E-ISSN 1433-0490
R&D Projects: GA ČR(CZ) GX19-27871X
Institutional support: RVO:67985840
Keywords : deterministic condensers * extractors * randomness * Santha-Vazirani sources
OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Impact factor: 0.582, year: 2020
Method of publishing: Limited access
https://doi.org/10.1007/s00224-020-09975-8

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 previousi − 1 bits is limited for every i ∈ [n]. If the dependence of the i’th bit on the remainingn − 1 bits is limited, then this is a strongSV-source. Even the strong SV -sources are known not to admit (universal) deterministic extractors, but they have seeded extractors, as their min-entropy is Ω 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 ε, δ ∈ (0,1) a deterministic condenser that maps n bits coming from a strong SV -source with bias at most δ to Ω n bits of min-entropy rate at least 1 − ε. 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.
Permanent Link: http://hdl.handle.net/11104/0309977