Pseudorandom bits for constant depth circuits


For every integerd we explicitly construct a family of functions (pseudo-random bit generators) that convert a polylogarithmic number of truly random bits ton bits that appear random to any family of circuits of polynomial size and depthd. The functions we construct are computable by a uniform family of circuits of polynomial size and constant depth. This allows us to simulate randomized constant depth polynomial size circuits inDSPACE(polylog) and inDTIME(2polylog). As a corollary we show that the complexity class AM is equal to the class of languages recognizable in NP with a random oracle. Our technique may be applied in order to get pseudo random generators for other complexity classes as well; a further paper [16] explores these issues.

This is a preview of subscription content, access via your institution.


  1. [1]

    L. Adleman: Two theorems on random polynomial time,19th FOCS pp. 75–83, 1978.

  2. [2]

    M. Ajtai: ∑ 11 formulas on finite structures.Annals of Pure and Applied Logic 24, pp. 1–48. 1983.

    Google Scholar 

  3. [3]

    M. Ajtai, andM. Ben-Or: A theorem on probabilistic constant depth computations,16th STOC, pp. 571–474, 1984.

  4. [4]

    M. Ajtai, andA. Wigderson: Deterministic simulation of probabilistic constant depth circuits26th FOCS, pp. 11–19, 1985.

  5. [5]

    L. Babai: Trading group theory for randomness17th STOC, pp. 421–429, 1975.

  6. [6]

    C. H. Benett, andJ. Gill: Relative to a random oracleA, P A ≠NP A ≠Co-NP A with probability 1.SIAM J. Comp. 10, 1981.

  7. [7]

    L. Babai, andS. Moran: Arthur Merlin games: a randomized proof system, and a hierarchy of complexity classes,J. Computer Sys. Sci. 36, pp. 254–276, 1988.

    Google Scholar 

  8. [8]

    M. Blum, andS. Micali: How to generate cryptographically strong sequences of pseudo random bits.23rd FOCS, pp. 112–117, 1982.

  9. [9]

    A. Chandra, D. Kozen, andL. Stockmeyer: Alternation,J. ACM,28, 1981.

  10. [10]

    M. Furst, R. J. Lipton, andL. Stoclmeyer: Pseudo random number generation and space complexity,Information and Control,64, 1985.

  11. [11]

    M. Furst, J. Saxe, andM. Sipser: Parity, Circuits, and the polynomial time hierarchy,22nd FOCS, pp. 260–270, 1981.

  12. [12]

    S. Goldwasser, andS. Micali: Probabilistic Encryption,JCSS,28, No. 2, 1984.

  13. [13]

    S. Goldwasser, andM. Sipser: Private coins vs. Public voins in interactive proof systems,18th STOC, pp. 59–68, 1986.

  14. [14]

    J. Hastad:Lower Bounds for the Size of Parity Circuits, Ph.D. Thesis, M.I.T., 1987.

  15. [15]

    S. A. Kurts: A note on randomized polynomial time,SIAM J. Comp. 16, No. 5, 1987.

    Google Scholar 

  16. [16]

    N. Nisan, andA. Wigderson: Hardness vs. Randomness,29th FOCS, 1988.

  17. [17]

    J. H. Reif, andJ. D. Tygar: Towards a theory of parallel randomized computation,TR-07-84, Aiken Computation Lab., Harvard University, 1984.

  18. [18]

    M. Sipser: A complexity theoretic approach to randomness,15th STOC, 330–335, 1983.

  19. [19]

    M. Sipser: Expanders, Randomness, or Time vs. Space, Structure in Complexity Theory, Lecture notes in Computer Science, No. 223, Ed. G. Goos, J. Hartmanis, pp. 325–329.

  20. [20]

    L. Stockmeyer: The polynomial time hierarchy,Theory. Comp. Sci. 3, No. 1, 1976.

  21. [21]

    A. C. Yao: Theory and applications of trapdoor functions,23rd FOCS, pp. 80–91, 1982.

  22. [22]

    A. C. Yao: Separating the polynomial time hierarchy by oracles,26th FOCS, pp. 1–10, 1985.

Download references

Author information



Additional information

Part of this work was done while the first author was in U. C. Berkeley, visiting the Hebrew University of Jerusalem.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

AMS subject classification (1980)

  • 68 C 25