Pseudorandom bits for constant depth circuits
 Noam Nisan
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For every integerd we explicitly construct a family of functions (pseudorandom 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(2^{ polylog }). 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.
 L. Adleman: Two theorems on random polynomial time,19th FOCS pp. 75–83, 1978.
 Ajtai, M. (1983) Annals of Pure and Applied Logic 24: pp. 148
 M. Ajtai, andM. BenOr: A theorem on probabilistic constant depth computations,16th STOC, pp. 571–474, 1984.
 M. Ajtai, andA. Wigderson: Deterministic simulation of probabilistic constant depth circuits26th FOCS, pp. 11–19, 1985.
 L. Babai: Trading group theory for randomness17th STOC, pp. 421–429, 1975.
 C. H. Benett, andJ. Gill: Relative to a random oracleA, P ^{ A } ≠NP ^{ A } ≠CoNP ^{ A } with probability 1.SIAM J. Comp. 10, 1981.
 Babai, L., Moran, S. (1988) Arthur Merlin games: a randomized proof system, and a hierarchy of complexity classes. J. Computer Sys. Sci. 36: pp. 254276
 M. Blum, andS. Micali: How to generate cryptographically strong sequences of pseudo random bits.23rd FOCS, pp. 112–117, 1982.
 A. Chandra, D. Kozen, andL. Stockmeyer: Alternation,J. ACM,28, 1981.
 M. Furst, R. J. Lipton, andL. Stoclmeyer: Pseudo random number generation and space complexity,Information and Control,64, 1985.
 M. Furst, J. Saxe, andM. Sipser: Parity, Circuits, and the polynomial time hierarchy,22nd FOCS, pp. 260–270, 1981.
 S. Goldwasser, andS. Micali: Probabilistic Encryption,JCSS,28, No. 2, 1984.
 S. Goldwasser, andM. Sipser: Private coins vs. Public voins in interactive proof systems,18th STOC, pp. 59–68, 1986.
 J. Hastad:Lower Bounds for the Size of Parity Circuits, Ph.D. Thesis, M.I.T., 1987.
 S. A. Kurts: A note on randomized polynomial time,SIAM J. Comp. 16, No. 5, 1987.
 N. Nisan, andA. Wigderson: Hardness vs. Randomness,29th FOCS, 1988.
 J. H. Reif, andJ. D. Tygar: Towards a theory of parallel randomized computation,TR0784, Aiken Computation Lab., Harvard University, 1984.
 M. Sipser: A complexity theoretic approach to randomness,15th STOC, 330–335, 1983.
 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.
 L. Stockmeyer: The polynomial time hierarchy,Theory. Comp. Sci. 3, No. 1, 1976.
 A. C. Yao: Theory and applications of trapdoor functions,23rd FOCS, pp. 80–91, 1982.
 A. C. Yao: Separating the polynomial time hierarchy by oracles,26th FOCS, pp. 1–10, 1985.
 Title
 Pseudorandom bits for constant depth circuits
 Journal

Combinatorica
Volume 11, Issue 1 , pp 6370
 Cover Date
 19910301
 DOI
 10.1007/BF01375474
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 68 C 25
 Industry Sectors
 Authors

 Noam Nisan ^{(1)}
 Author Affiliations

 1. Department of Computer Science, Hebrew University of Jerusalem, Israel