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Deterministic Randomness Extraction from Generalized and Distributed Santha-Vazirani Sources

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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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 Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.

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Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

    Salman Beigi, Omid Etesami & Amin Gohari

  2. Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

    Amin Gohari

Authors
  1. Salman Beigi
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  2. Omid Etesami
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  3. Amin Gohari
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Corresponding author

Correspondence to Omid Etesami .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Beigi, S., Etesami, O., Gohari, A. (2015). Deterministic Randomness Extraction from Generalized and Distributed Santha-Vazirani Sources. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_12

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47671-0

  • Online ISBN: 978-3-662-47672-7

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