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)


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Goldreich, O.: Simple constructions of almost k-wise independent random variables. In: FOCS 1990, pp. 544–553 (1990)Google Scholar
  2. 2.
    Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7(4), 567–583 (1986)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barak, B., Impagliazzo, R., Wigderson, A.: Extracting randomness using few independent sources. In: FOCS 2004, pp. 384–393 (2004)Google Scholar
  4. 4.
    Barak, B., Kindler, G., Shaltiel, R., Sudakov, B., Wigderson, A.: Simulating Independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors. In: STOC 2005, pp. 1–10 (2005)Google Scholar
  5. 5.
    Bourgain, J.: More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory 1(1), 1–32 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chor, B., Goldreich, O.: Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17(2), 230–261 (1988)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chor, B., Goldreich, O., Håstad, J., Friedman, J., Rudich, S., Smolensky, R.: The bit extraction problem of t-resilient functions. In: FOCS 1985, pp. 396–407 (1985)Google Scholar
  8. 8.
    Dodis, Y., Elbaz, A., Oliveira, R., Raz, R.: Improved randomness extraction from two independent sources. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 334–344. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Gabizon, A., Raz, R., Shaltiel, R.: Deterministic extractors for bit-fixing sources by obtaining an independent seed. FOCS 2004, 394–403 (2004)Google Scholar
  10. 10.
    Impagliazzo, R., Shaltiel, R., Wigderson, A.: Extractors and pseudo-random generators with optimal seed length. In: STOC 2000, pp. 1–10 (2000)Google Scholar
  11. 11.
    Jukna, S.: Extremal Combinatorics. Springer, Heidelberg (2001)MATHGoogle Scholar
  12. 12.
    Kamp, J., Rao, A., Vahan, S., Zuckerman, D.: Deterministic extractors for small-space sources. In: STOC 2006 (2006)Google Scholar
  13. 13.
    Kamp, J., Zuckerman, D.: Deterministic extractors for bit-fixing sources and exposure-resilient cryptography. In: FOCS 2003, pp. 92–101 (2003)Google Scholar
  14. 14.
    König, R., Maurer, U.: Generalized strong extractors and deterministic privacy amplification. In: Proc. Cryptography and Coding, pp. 322–339 (2005)Google Scholar
  15. 15.
    Lee, C.-J., Lu, C.-J., Tsai, S.-C., Tzeng, W.-G.: Extracting randomness from multiple independent sources. IEEE Transactions on Information Theory 51(6), 2224–2227 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Lu, C.-J.: Encryption against storage-bounded adversaries from on-line strong extractors. J. Cryptology 17(1), 27–42 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lu, C.-J., Reingold, O., Vadhan, S., Wigderson, A.: Extractors: Optimal up to constant factors. In: STOC 2003, pp. 602–611 (2003)Google Scholar
  18. 18.
    Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Nisan, N., Ta-Shma, A.: Extracting randomness: A survey and new constructions. J. Comput. Syst. Sci. 58(1), 148–173 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Raz, R.: Extractors with weak random seeds. In: STOC 2005, pp. 11–20 (2005)Google Scholar
  22. 22.
    Raz, R., Reingold, O., Vadhan, S.: Extracting all the randomness and reducing the error in Trevisan’s extractors. In: STOC 1999, pp. 149–158 (1999)Google Scholar
  23. 23.
    Radhakrishnan, J., Ta-Shma, A.: Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM J. Discrete Math. 13(1), 2–24 (2000)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Reingold, O., Shaltiel, R., Wigderson, A.: Extracting randomness via repeated condensing. In: FOCS 2000, pp. 12–14 (2000)Google Scholar
  25. 25.
    Shaltiel, R.: Recent developments in explicit constructions of extractors. Bulletin of the European Association for Theoretical Computer Science 77, 67–95 (2002)MATHMathSciNetGoogle Scholar
  26. 26.
    Shaltiel, R., Umans, C.: Simple extractors for all min-entropies and a new pseudo-random generator. In: FOCS 2001, pp. 648–657 (2001)Google Scholar
  27. 27.
    Ta-Shma, A., Umans, C., Zuckerman, D.: Loss-less condensers, unbalanced expanders, and extractors. In: STOC 2001, pp. 143–152 (2001)Google Scholar
  28. 28.
    Ta-Shma, A., Zuckerman, D.: Extractor codes. STOC 2001, pp. 193–199 (2001)Google Scholar
  29. 29.
    Trevisan, L.: Extractors and pseudorandom generators. JACM 48(4), 860–879 (2001)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Vadhan, S.: Constructing locally computable extractors and cryptosystems in the bounded-storage model. J. Cryptology 17(1), 43–77 (2004)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wigderson, A., Zuckerman, D.: Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica 19(1), 125–138 (1999)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Zuckerman, D.: General weak random sources. In: FOCS 1990, pp. 534–543 (1990)Google Scholar
  33. 33.
    Zuckerman, D.: Simulating BPP using a general weak random source. Algorithmica 16(4/5), 367–391 (1996)MATHMathSciNetGoogle Scholar
  34. 34.
    Zuckerman, D.: Randomness-optimal oblivious sampling. Random Structures and Algorithms 11, 345–367 (1997)MATHCrossRefMathSciNetGoogle Scholar

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

Personalised recommendations