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Deterministic Extractors for Independent-Symbol Sources

  • Chia-Jung Lee
  • Chi-Jen Lu
  • Shi-Chun Tsai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from {0,1} d . The only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which can extract Ω(logk – logd – loglog(1/ε)) bits with error ε, for any n,d,k ∈ ℕ and ε∈(0,1). For sources with a larger min-entropy, we can extract even more randomness. When kn 1/2 + γ, for any constant γ∈(0,1/2), we can extract m=kO(d log(1/ε)) bits with any error \(\varepsilon \ge 2^{-\Omega(n^{\gamma})}\). When k≥log c n, for some constant c>0, we can extract m=kd (1/ε) O(1) bits with any error εk  − − Ω(1). Our results generalize those of Kamp & Zuckerman and Gabizon et al. which only work for bit-fixing sources (with d=1 and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a non-explicit deterministic extractor which can extract m=kO(log(1/ε)) bits whenever k=ω(d+log(n/ε)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss km = Ω(log(1/ε)). This generalizes a lower bound of Radhakrishnan & Ta-Shma with respect to general sources.

Keywords

Explicit Construction Independent Source Average Argument Entropy Loss Probabilistic Argument 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Chia-Jung Lee
    • 1
  • Chi-Jen Lu
    • 2
  • Shi-Chun Tsai
    • 1
  1. 1.Department of Computer ScienceNational Chiao-Tung UniversityHsinchuTaiwan
  2. 2.Institute of Information ScienceAcademia SinicaTaipeiTaiwan

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