Complexity bounds on general hard-core predicates

Abstract

A Boolean function t is a hard-core predicate for a one-way function f if b is polynomial-time computable but b(x) is difficult 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-ε)) average sensitivity, for any ε > 0. As a result, such families cannot consist of • functions in AC0, • monotone functions, • functions computed by generalized threshold gates, or • symmetric -threshold functions, ford = O(n1/2-ε) and ε > 0. © 2001 International Association for Cryptologic Research.