A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
The computational complexity of universal hashing
Theoretical Computer Science - Special issue on structure in complexity theory
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
A Pseudorandom Generator from any One-way Function
SIAM Journal on Computing
Modern Cryptography, Probabilistic Proofs, and Pseudorandomness
Modern Cryptography, Probabilistic Proofs, and Pseudorandomness
Pseudorandomness and Cryptographic Applications
Pseudorandomness and Cryptographic Applications
The Complexity of Computing Hard Core Predicates
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Universal hash functions & hard core bits
EUROCRYPT'95 Proceedings of the 14th annual international conference on Theory and application of cryptographic techniques
FCT '01 Proceedings of the 13th International Symposium on Fundamentals of Computation Theory
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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 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.