Pseudorandom bits for constant depth circuits

For every integer d we explicitly construct a family of functions (pseudo-random bit generators) that convert a polylogarithmic number of truly random bits to n bits that appear random to any family of circuits of polynomial size and depth d. 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 in DSPACE(polylog) and in DTIME(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.