Skip to main content
SpringerLink
Log in

An Introduction to Randomness Extractors

  • Conference paper
Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6756))

Included in the following conference series:

Abstract

We give an introduction to the area of “randomness extraction” and survey the main concepts of this area: deterministic extractors, seeded extractors and multiple sources extractors. For each one we briefly discuss background, definitions, explicit constructions and applications.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  2. Ajtai, M., Komlós, J., Szemerédi, E.: Deterministic simulation in logspace. In: STOC, pp. 132–140 (1987)

    Google Scholar 

  3. Alon, N.: The shannon capacity of a union. Combinatorica 18(3), 301–310 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barak, B., Impagliazzo, R., Wigderson, A.: Extracting randomness using few independent sources. SIAM J. Comput. 36(4), 1095–1118 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barak, B., Kindler, G., Shaltiel, R., Sudakov, B., Wigderson, A.: Simulating independence: New constructions of condensers, ramsey graphs, dispersers, and extractors. J. ACM 57(4) (2010)

    Google Scholar 

  6. Bourgain, J., Katz, N., Tao, T.: A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14(1), 27–57 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Blum, M.: Independent unbiased coin flips from a correlated biased source-a finite stae markov chain. Combinatorica 6(2), 97–108 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bourgain, J.: More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory 1, 1–32 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bourgain, J.: On the construction of affine extractors. Geometric And Functional Analysis 17(1), 33–57 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Barak, B., Rao, A., Shaltiel, R., Wigderson, A.: 2-source dispersers for sub-polynomial entropy and ramsey graphs beating the frankl-wilson construction. In: STOC, pp. 671–680 (2006)

    Google Scholar 

  11. Ben-Sasson, E., Kopparty, S.: Affine dispersers from subspace polynomials. In: STOC, pp. 65–74 (2009)

    Google Scholar 

  12. Ben-Sasson, E., Zewi, N.: From affine to two-source extractors via approximate duality. In: STOC (2011)

    Google Scholar 

  13. Chor, B., Goldreich, O.: Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing 17(2), 230–261 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chor, B., Goldreich, O., Håstad, J., Friedman, J., Rudich, S., Smolensky, R.: The bit extraction problem of t-resilient functions (preliminary version). In: FOCS, pp. 396–407 (1985)

    Google Scholar 

  15. Capalbo, M.R., Reingold, O., Vadhan, S.P., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: STOC, pp. 659–668 (2002)

    Google Scholar 

  16. DeVos, M., Gabizon, A.: Simple affine extractors using dimension expansion. In: IEEE Conference on Computational Complexity, pp. 50–57 (2010)

    Google Scholar 

  17. Dvir, Z., Gabizon, A., Wigderson, A.: Extractors and rank extractors for polynomial sources. Computational Complexity 18(1), 1–58 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dvir, Z., Kopparty, S., Saraf, S., Sudan, M.: Extensions to the method of multiplicities, with applications to kakeya sets and mergers. In: FOCS, pp. 181–190 (2009)

    Google Scholar 

  19. Dvir, Z.: Extractors for varieties. In: IEEE Conference on Computational Complexity, pp. 102–113 (2009)

    Google Scholar 

  20. Elias, P.: The efficient construction of an unbiased random sequence. Ann. Math. Statist. 43, 865–870 (1972)

    Article  MATH  Google Scholar 

  21. Frankl, P., Wilson, R.M.: Intersection theorems with geometric consequences. Combinatorica 1(4), 357–368 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gillman, D.: A chernoff bound for random walks on expander graphs. SIAM J. Comput. 27(4), 1203–1220 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gopalan, P.: Constructing ramsey graphs from boolean function representations. In: IEEE Conference on Computational Complexity, pp. 115–128 (2006)

    Google Scholar 

  24. Gabizon, A., Raz, R.: Deterministic extractors for affine sources over large fields. Combinatorica 28(4), 415–440 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Grolmusz, V.: Superpolynomial size set-systems with restricted intersections mod 6 and explicit ramsey graphs. Combinatorica 20(1), 71–86 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gabizon, A., Raz, R., Shaltiel, R.: Deterministic extractors for bit-fixing sources by obtaining an independent seed. SIAM J. Comput. 36(4), 1072–1094 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gabizon, A., Shaltiel, R.: Increasing the output length of zero-error dispersers. In: APPROX-RANDOM, pp. 430–443 (2008)

    Google Scholar 

  28. Guruswami, V.: Algorithmic results in list decoding. Foundations and Trends in Theoretical Computer Science 2(2) (2007)

    Google Scholar 

  29. Guruswami, V., Umans, C., Vadhan, S.P.: Unbalanced expanders and randomness extractors from parvaresh–vardy codes. J. ACM 56(4) (2009)

    Google Scholar 

  30. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Amer. Math. Soc. 43, 439–561 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kahale, N.: Eigenvalues and expansion of regular graphs. J. Assoc. Comput. Mach. 42, 1091–1106 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kalai, Y.T., Li, X., Rao, A.: 2-source extractors under computational assumptions and cryptography with defective randomness. In: FOCS, pp. 617–626 (2009)

    Google Scholar 

  33. Kalai, Y.T., Li, X., Rao, A., Zuckerman, D.: Network extractor protocols. In: FOCS, pp. 654–663 (2008)

    Google Scholar 

  34. König, R., Maurer, U.M.: Generalized strong extractors and deterministic privacy amplification. In: IMA Int. Conf., pp. 322–339 (2005)

    Google Scholar 

  35. Konyagin, S.V.: A sum-product estimate in fields of prime order. Arxiv technical report (2003), http://arxiv.org/abs/math.NT/0304217

  36. Kamp, J., Rao, A., Vadhan, S.P., Zuckerman, D.: Deterministic extractors for small-space sources. J. Comput. Syst. Sci. 77(1), 191–220 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kinne, J., van Melkebeek, D., Shaltiel, R.: Pseudorandom generators and typically-correct derandomization. In: APPROX-RANDOM, pp. 574–587 (2009)

    Google Scholar 

  38. Kamp, J., Zuckerman, D.: Deterministic extractors for bit-fixing sources and exposure-resilient cryptography. SIAM J. Comput. 36(5), 1231–1247 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Li, X.: Improved constructions of three source extractors. In: IEEE Conference on Computational Complexity (2011)

    Google Scholar 

  40. Li, X.: A new approach to affine extractors and dispersers. In: IEEE Conference on Computational Complexity (2011)

    Google Scholar 

  41. Lu, C.-J., Reingold, O., Vadhan, S.P., Wigderson, A.: Extractors: optimal up to constant factors. In: STOC, pp. 602–611 (2003)

    Google Scholar 

  42. Nisan, N., Ta-Shma, A.: Extracting randomness: A survey and new constructions. J. Comput. Syst. Sci. 58(1), 148–173 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  43. Nisan, N., Wigderson, A.: Hardness vs randomness. J. Comput. Syst. Sci. 49(2), 149–167 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  44. Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  45. Peres, Y.: Iterating von neumann’s procedure for extracting random bits. Ann. Statist. 20, 590–597 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  46. Parvaresh, F., Vardy, A.: Correcting errors beyond the guruswami-sudan radius in polynomial time. In: FOCS, pp. 285–294 (2005)

    Google Scholar 

  47. Rao, A.: An exposition of bourgain’s 2-source extractor. Electronic Colloquium on Computational Complexity (ECCC) 14(034) (2007)

    Google Scholar 

  48. Rao, A.: Extractors for a constant number of polynomially small min-entropy independent sources. SIAM J. Comput. 39(1), 168–194 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  49. Rao, A.: Extractors for low-weight affine sources. In: IEEE Conference on Computational Complexity, pp. 95–101 (2009)

    Google Scholar 

  50. Raz, R.: Extractors with weak random seeds. In: STOC, pp. 11–20 (2005)

    Google Scholar 

  51. Raz, R., Reingold, O., Vadhan, S.P.: Extracting all the randomness and reducing the error in trevisan’s extractors. J. Comput. Syst. Sci. 65(1), 97–128 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  52. Radhakrishnan, J., Ta-Shma, A.: Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics 13(1), 2–24 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  53. Reingold, O., Vadhan, S.P., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In: FOCS, pp. 3–13 (2000)

    Google Scholar 

  54. Rao, A., Zuckerman, D.: Extractors for three uneven-length sources. In: APPROX-RANDOM, pp. 557–570 (2008)

    Google Scholar 

  55. Shaltiel, R.: Recent developments in explicit constructions of extractors. Bulletin of the EATCS 77, 67–95 (2002); Special issue on cryptography

    MATH  MathSciNet  Google Scholar 

  56. Shaltiel, R.: How to get more mileage from randomness extractors. Random Struct. Algorithms 33(2), 157–186 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  57. Shaltiel, R.: Weak derandomization of weak algorithms: Explicit versions of yao’s lemma. In: IEEE Conference on Computational Complexity, pp. 114–125 (2009)

    Google Scholar 

  58. Shaltiel, R.: Dispersers for affine sources with sub-polynomial entropy (unpublished, 2011)

    Google Scholar 

  59. Saks, M.E., Srinivasan, A., Zhou, S.: Explicit or-dispersers with polylogarithmic degree. J. ACM 45(1), 123–154 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  60. Shaltiel, R., Umans, C.: Simple extractors for all min-entropies and a new pseudorandom generator. J. ACM 52(2), 172–216 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  61. Santha, M., Vazirani, U.V.: Generating quasi-random sequences from semi-random sources. J. Comput. Syst. Sci. 33(1), 75–87 (1986)

    Article  MATH  Google Scholar 

  62. Trevisan, L.: Extractors and pseudorandom generators. J. ACM 48(4), 860–879 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  63. Ta-Shma, A.: Almost optimal dispersers. Combinatorica 22(1), 123–145 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  64. Ta-Shma, A., Umans, C., Zuckerman, D.: Lossless condensers, unbalanced expanders, and extractors. Combinatorica 27(2), 213–240 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  65. Ta-Shma, A., Zuckerman, D.: Extractor codes. IEEE Transactions on Information Theory 50(12), 3015–3025 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  66. Ta-Shma, A., Zuckerman, D., Safra, S.: Extractors from reed-muller codes. J. Comput. Syst. Sci. 72(5), 786–812 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  67. Trevisan, L., Vadhan, S.P.: Extracting randomness from samplable distributions. In: FOCS, pp. 32–42 (2000)

    Google Scholar 

  68. Vadhan, S.P.: The unified theory of pseudorandomness. SIGACT News 38(3), 39–54 (2007)

    Article  Google Scholar 

  69. Vazirani, U.V.: Efficiency considerations in using semi-random sources (extended abstract). In: STOC, pp. 160–168 (1987)

    Google Scholar 

  70. von Neumann, J.: Various techniques used in connection with random digits. Applied Math Series 12, 36–38 (1951)

    Google Scholar 

  71. Vazirani, U.V., Vazirani, V.V.: Random polynomial time is equal to slightly-random polynomial time. In: FOCS, pp. 417–428 (1985)

    Google Scholar 

  72. Wigderson, A., Zuckerman, D.: Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica 19(1), 125–138 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  73. Yehudayoff, A.: Affine extractors over prime fields (2010) (manuscript)

    Google Scholar 

  74. Zuckerman, D.: Randomness-optimal oblivious sampling. Random Struct. Algorithms 11(4), 345–367 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Haifa, Israel

    Ronen Shaltiel

Authors
  1. Ronen Shaltiel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Shaltiel, R. (2011). An Introduction to Randomness Extractors. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6756. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22012-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22012-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22011-1

  • Online ISBN: 978-3-642-22012-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation