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Deterministic randomness extraction from generalized and distributed Santha-Vazirani sources

Beigi, S ; Sharif University of Technology | 2017

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  1. Type of Document: Article
  2. DOI: 10.1137/15M1027206
  3. Publisher: Society for Industrial and Applied Mathematics Publications , 2017
  4. Abstract:
  5. 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 nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to s "nondegenerate" dice that are c-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if s < c. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs (a1; b1); (a2; b2),... distributed between two parties, where the first party receives ai's and the second one receives bi'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 (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources. © 2017 Society for Industrial and Applied Mathematics
  6. Keywords:
  7. Common randomness extraction ; Randomness extraction ; Santha-Vazirani sources ; Extraction ; Common randomness ; Conditional distribution ; Independently and identically distributed ; Maximal correlation ; Nonbinary sequences ; Nondegenerate ; Random sequence ; Randomness extractions ; Random processes
  8. Source: SIAM Journal on Computing ; Volume 46, Issue 1 , 2017 , Pages 1-36 ; 00975397 (ISSN)
  9. URL: https://epubs.siam.org/doi/10.1137/15M1027206