Using hard problems to derandomize algorithms: An incomplete survey

  • Russell Impagliazzo
Invited Talk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)


Yao showed how to use a sufficiently secure cryptographic permutation to construct pseudo-random generators to de-randomize arbitrary randomized algorithms. To do this, he used the fact that the XOR of independent random instances of a somewhat hard Boolean problem becomes almost completely unpredictable, a “direct product lemma”. In this survey, we try to sketch various connections between hard problems, direct product results, and de-randomization of algorithms.


Boolean Function Hard Problem Random Input Acceptance Probability Probabilistic Algorithm 
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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  1. 1.
    A. Andreev, A. Clementi and J. Rolim, “Hitting Sets Derandomize BPP”, in XXIII International Colloquium on Algorithms, Logic and Programming (ICALP'96), 1996.Google Scholar
  2. 2.
    A. Andreev, A. Clementi, and J. Rolim, “Hitting Properties of Hard Boolean Operators and its Consequences on BPP”, manuscript, 1996.Google Scholar
  3. 3.
    L. Babai, L. Fortnow, N. Nisan and A. Wigderson, “BPP has Subexponential Time Simulations unless EXPTIME has Publishable Proofs”, Complexity Theory, Vol 3, pp. 307–318, 1993.Google Scholar
  4. 4.
    M. Blum and S. Micali. “How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits”, SIAM J. Comput., Vol. 13, pages 850–864, 1984.Google Scholar
  5. 5.
    J. Feigenbaum and L. Fortnow, “Random self-reducibility of complete sets”, SIAM J. of Computing, vol. 22, no. 5 (Oct. 1993). pp. 995–1005.Google Scholar
  6. 6.
    O. Goldreich and L.A. Levin. “A Hard-Core Predicate for all One-Way Functions”, in ACM Symp. on Theory of Computing, pp. 25–32, 1989.Google Scholar
  7. 7.
    [GNW] O. Goldreich, N. Nisan and A. Wigderson. “On Yao's XOR-Lemma”, available via www at ECCC TR95-050, 1995.Google Scholar
  8. 8.
    J. Hastad, R. Impagliazzo, L.A. Levin and M. Luby, “Construction of Pseudorandom Generator from any One-Way Function”, to appear in SICOMP. (See preliminary versions by Impagliazzo et. al. in 21st STOC and Hastad in 22nd STOC).Google Scholar
  9. 9.
    R. Impagliazzo, “Hard-core Distributions for Somewhat Hard Problems”, in 36th FOCS, pages 538–545, 1995.Google Scholar
  10. 10.
    R. Impagliazzo and A. Wigderson, “P=BPP Unless E has Sub-exponential circuits: De-randomizing the XOR Lemma”, STOC '97, pp. 220–229.Google Scholar
  11. 11.
    L.A. Levin, “One-Way Functions and Pseudorandom Generators”, Combinatorica, Vol. 7, No. 4, pp. 357–363, 1987.Google Scholar
  12. 12.
    N. Nisan, “Pseudo-random bits for constant depth circuits”, Combinatorica 11 (1), pp. 63–70, 1991.Google Scholar
  13. 13.
    N. Nisan, and A. Wigderson, “Hardness vs Randomness”, J. Comput. System Sci. 49, 149–167, 1994Google Scholar
  14. 14.
    S. Rudich, “WP-Natural Proofs”, this conference, to appear.Google Scholar
  15. 15.
    M. Sipser, “A complexity-theoretic approach to randomness”. STOC '83, pp. 330–335.Google Scholar
  16. 16.
    A.C. Yao, “Theory and Application of Trapdoor Functions”, in 23st FOCS, pages 80–91, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Russell Impagliazzo
    • 1
  1. 1.Computer Science and Engineering, UC, San DiegoLa Jolla

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