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  explores these issues.
AMS subject classification (1980)68 C 25
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