Spectral 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(n ^1−∈) 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(n ^1/2−∈) and ∈ > 0. This also subsumes a 1997 result of Goldmann and Näslund 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.

Part of this work was done while visiting McGill University

Supported by NSF NYI Grant No. CCR-9457799 and a David and Lucile Packard Fellowship for Science and Engineering. This research was done while the author was a postdoc at the University of Texas at Austin.