Spectral Bounds on General Hard Core Predicates

A Boolean function b is a hard core predicate for a one-way function f if b is polynomial time computable but b(x) is diffcult to predict from f(x). A general family of hard core predicates is a family of functions containing a hard core predicate for any one-way function. A seminal result of Goldreich and Levin asserts that the family of parity functions is a general family of hard core predicates. We show that no general family of hard core predicates can consist of functions with O(n1-∈) average sensitivity, for any ∈ > 0. As a result, such families cannot consist of monotone functions, functions computed by generalized threshold gates, or symmetric d-threshold functions, for d = O(n1/2-∈) and ∈ > 0. This also subsumes a 1997 result of Goldmann and Naslund which asserts that such families cannot consist of functions computable in AC0. The above bound on sensitivity is obtained by (lower) bounding the high order terms of the Fourier transform.