The Complexity of Computing Hard Core Predicates

We prove that a general family of hard core predicates requires circuits of depth (l-o(l)) ~ or super-polynomial size to be realized. This lower bound is essentially tight, l~or constant depth circuits, an exponential lower bound on the size is obtained. Assuming the existence of one-way functions, we explicitly construct a one-way function f(x) such that for any circuit c from a family of circuits as above, c(x) is almost always predictable from f(x).