Complexity Bounds on General Hard-Core Predicates

Abstract. 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 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 d -threshold functions, for d = O(n1/2 - ε) and ε > 0 .

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