skip to main content
research-article

Kolmogorov Complexity in Randomness Extraction

Published:01 August 2011Publication History
Skip Abstract Section

Abstract

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction and randomness extraction. We present a distribution Mk based on Kolmogorov complexity that is complete for randomness extraction in the sense that a computable function is an almost randomness extractor if and only if it extracts randomness from Mk.

References

  1. Barak, B., Impagliazzo, R., and Wigderson, A. 2006. Extracting randomness using few independent sources. SIAM J. Comput. 36, 4, 1095--1118. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Bourgain, J. 2005. More on the sum-product phenomena in prime fields and its applications. Int. J. Number Theor. 1, 1--32.Google ScholarGoogle ScholarCross RefCross Ref
  3. Buhrman, H., Fortnow, L., Newman, I., and Vereshchagin, N. 2005. Increasing Kolmogorov complexity. In Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 3404. Springer, 412--421. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Chor, B. and Goldreich, O. 1988. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 2, 230--261. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Fortnow, L., Hitchcock, J. M., Pavan, A., Vinodchandran, N. V., and Wang, F. 2011. Extracting Kolmogorov complexity with applications to dimension zero-one laws. Inf. Comput. 209, 4, 627--636. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Guruswami, V., Umans, C., and Vadhan, S. P. 2009. Unbalanced expanders and randomness extractors from parvaresh--vardy codes. J. ACM 56, 4. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Kalai, Y. T., Li, X., and Rao, A. 2009. 2-source extractors under computational assumptions and cryptography with defective randomness. In Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 617--626. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Li, M. and Vitányi, P. 1991. Learning simple concepts under simple distributions. SIAM J. Comput. 20, 5, 911--935. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Li, M. and Vitányi, P. 1992. Average case complexity under the universal distribution equals worst-case complexity. Inf. Process. Lett. 42, 3, 145--149. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Li, M. and Vitányi, P. 1997. An Introduction to Kolmogorov Complexity and Its Applications. Springer. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Nisan, N. and Ta-Shma, A. 1999. Extracting randomness: A survey and new constructions. J. Comput. Syst. Sci. 42, 2, 149--167. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Rao, A. 2006. Randomness extractors for independent sources and applications. Ph.D. thesis, University of Texas, Austin. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Rao, A. 2008. A 2-source almost-extractor for linear entropy. In Proceedings of the APPROX-RANDOM Conference. Lecture Notes in Computer Science. Springer, 549--556. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Shaltiel, R. 2004. Current Trends in Theoretical Computer Science. vol 1 Algorithms and Complexity. World Scientific (Chapter Recent Developments in Extractors).Google ScholarGoogle Scholar
  15. Zimand, M. 2009. Extracting the Kolmogorov complexity of strings and sequences from sources with limited independence. In Proceedings of the Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science. Springer, 607--708.Google ScholarGoogle Scholar
  16. Zimand, M. 2010. Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences. Theory Comput. Syst. 46, 4, 707--722. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Kolmogorov Complexity in Randomness Extraction

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 3, Issue 1
      August 2011
      35 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2003685
      Issue’s Table of Contents

      Copyright © 2011 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 August 2011
      • Revised: 1 May 2011
      • Accepted: 1 May 2011
      • Received: 1 November 2010
      Published in toct Volume 3, Issue 1

      Permissions

      Request permissions about this article.

      Request Permissions

      Check for updates

      Qualifiers

      • research-article
      • Research
      • Refereed

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!