Skip to main content

Pseudorandomness

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

Abstract

We postulate that a distribution is pseudorandom if it cannot be told apart from the uniform distribution by any efficient procedure. This yields a robust definition of pseudorandom generators as efficient deterministic programs stretching short random seeds into longer pseudorandom sequences. Thus, pseudorandom generators can be used to reduce the randomness-complexity in any efficient procedure. Pseudorandom generators and computational difficulty are closely related: loosely speaking, each can be efficiently transformed into the other.

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

Buying options

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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai, J. Komlos, E. Szemeredi. Deterministic Simulation in LogSpace. In 19th ACM Symposium on the Theory of Computing, pages 132–140, 1987.

    Google Scholar 

  2. N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567–583, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Bellare, O. Goldreich, and S. Goldwasser. Randomness in Interactive Proofs. Computational Complexity, Vol. 4, No. 4, pages 319–354, 1993.

    Article  MathSciNet  Google Scholar 

  4. M. Blum and S. Micali. How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. SIAM Journal on Computing, Vol. 13, pages 850–864, 1984. Preliminary version in 23rd IEEE Symposium on Foundations of Computer Science, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Carter and M. Wegman. Universal Hash Functions. Journal of Computer and System Science, Vol. 18, 1979, pages 143–154.

    Article  MATH  MathSciNet  Google Scholar 

  6. GJ. Chaitin. On the Length of Programs for Computing Finite Binary Sequences. Journal of the ACM, Vol. 13, pages 547–570, 1966.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Chor and O. Goldreich. On the Power ofTwo-Point Based Sampling. Jour, of Complexity, Vol 5, 1989, pages 96–106. Preliminary version dates 1985.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Chor and O. Goldreich. Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity. SIAM Journal on Computing, Vol. 17, No. 2, pages 230–261, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  9. T.M. Cover and G.A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc., New-York, 1991.

    MATH  Google Scholar 

  10. O. Goldreich. Foundationof Cryptography-Fragments of a Book. February 1995. Available from http://theory.lcs.mit.edu/~oded/frag.html.

  11. O. Goldreich. Modern Cryptography, Probabilistic Proofs and Pseudorandomness Algorithms and Combinatorics series (Vol. 17), Springer, 1998.

    Google Scholar 

  12. O. Goldreich, S. Goldwasser, and S. Micali. How to Construct Random Functions. Journal of the ACM, Vol. 33, No. 4, pages 792–807, 1986.

    Article  MathSciNet  Google Scholar 

  13. O. Goldreich and L.A. Levin. Hard-core Predicates for any One-Way Function. In 21st ACM Symposium on the Theory of Computing, pages 25–32, 1989.

    Google Scholar 

  14. O. Goldreich and S. Micali. Increasing the Expansion of Pseudorandom Generators. Unpublished manuscript, 1984.

    Google Scholar 

  15. O. Goldreich, and H. Krawczyk. On Sparse Pseudorandom Ensembles. Random Structures and Algorithms, Vol. 3, No. 2, (1992), pages 163–174.

    Article  MATH  MathSciNet  Google Scholar 

  16. O. Goldreich, H. Krawcyzk and M. Luby. On the Existence of Pseudorandom Generators. SIAM Journal on Computing, Vol. 22–6, pages 1163–1175, 1993.

    Article  Google Scholar 

  17. S. Goldwasser and S. Micali. Probabilistic Encryption. Journal of Computer and System Science, Vol. 28, No. 2, pages 270–299, 1984. Preliminary version in 14th ACM Symposium on the Theory of Computing, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Hastad, R. Impagliazzo, L.A. Levin and M. Luby. A Pseudorandom Generator from any One-way Function. SIAM Journal on Computing, Volume 28, Number 4, pages 1364–1396, 1999. Preliminary versions by Impagliazzo et. al. in 21st ACM Symposium on the Theory of Computing (1989) and Hastad in 22nd ACM Symposium on the Theory of Computing (1990).

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Impagliazzo and A. Wigderson. P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma. In 29th ACM Symposium on the Theory of Computing, pages 220–229, 1997.

    Google Scholar 

  20. D.E. Knuth. The Art of Computer Programming, Vol. 2 (Seminumerical Algorithms). Addison-Wesley Publishing Company, Inc., 1969 (first edition) and 1981 (second edition).

    Google Scholar 

  21. A. Kolmogorov. Three Approaches to the Concept of “The Amount Of Information”. Probl. of Inform. Transm., Vol. 1/1, 1965.

    Google Scholar 

  22. L. A. Levin. Randomness Conservation Inequalities: Information and Independence in Mathematical Theories. Inform, and Control, Vol. 61, pages 15–37, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Li and P. Vitanyi. An Introduction to Kolmogorov Complexity and its Applications. Springer Verlag, August 1993.

    Google Scholar 

  24. J. Naor and M. Naor. Small-bias Probability Spaces: Efficient Constructions and Applications. SIAMJ. on Computing, Vol 22, 1993, pages 838–856.

    Article  MATH  MathSciNet  Google Scholar 

  25. N. Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, Vol. 11(1), pages 63–70, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. Nisan. Pseudorandom Generators for Space Bounded Computation. Combinatorica, Vol. 12(4), pages 449–461, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  27. N. Nisan. RL ⊆SC. Journal of Computational Complexity, Vol. 4, pages 1–11, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  28. N. Nisan and A. Wigderson. Hardness vs Randomness. Journal of Computer and System Science, Vol. 49, No. 2, pages 149–167, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  29. N. Nisan and D. Zuckerman. Randomness is Linear in Space. Journal of Computer and System Science, Vol. 52(1), pages 43–52, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  30. A.M. Odlyzko. The future of integer factorization. CryptoBytes (The technical newsletter of RSA Laboratories), Vol. 1 (No. 2), pages 5–12, 1995. Available from http://www.research.att.com/~amo

    Google Scholar 

  31. A.M. Odlyzko. Discrete logarithms and smooth polynomials. In Finite Fields: Theory, Applications and Algorithms, G. L. Mullen and P. Shiue, eds., Amer. Math. Soc, Contemporary Math. Vol. 168, pages 269–278, 1994. Available from http://www.research.att.com/~amo

  32. A.R. Razborov and S. Rudich. Natural proofs. Journal of Computer and System Science, Vol. 55(1), pages 24–35, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  33. C.E. Shannon. A mathematical theory of communication. Bell Sys. Tech. Jour, Vol. 27, pages 623–656, 1948.

    MathSciNet  Google Scholar 

  34. RJ. Solomonoff. A Formal Theory of Inductive Inference. Inform, and Control, Vol. 7/1, pages 1–22, 1964.

    Article  MathSciNet  Google Scholar 

  35. L. Valiant. A theory of the learnable. Communications of the ACM, Vol. 27/11, pages 1134–1142, 1984.

    Article  Google Scholar 

  36. L. Trevisan. Constructions of Near-Optimal Extractors Using Pseudo-Random Generators. In 31st ACM Symposium on the Theory of Computing, pages 141–148, 1998.

    Google Scholar 

  37. A.C. Yao. Theory and Application of Trapdoor Functions. In 23rd IEEE Symposium on Foundations of Computer Science, pages 80–91, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Goldreich, O. (2000). Pseudorandomness. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_58

Download citation

  • DOI: https://doi.org/10.1007/3-540-45022-X_58

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics