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A hard-core predicate for all one-way functions

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L. A. LevinAuthors Info & Claims

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

Pages 25 - 32

https://doi.org/10.1145/73007.73010

Published: 01 February 1989 Publication History

Abstract

A central tool in constructing pseudorandom generators, secure encryption functions, and in other areas are “hard-core” predicates b of functions (permutations) ƒ, discovered in [Blum Micali 82]. Such b(x) cannot be efficiently guessed (substantially better than 50-50) given only ƒ(x). Both b, ƒ are computable in polynomial time.

[Yao 82] transforms any one-way function ƒ into a more complicated one, ƒ^*, which has a hard-core predicate. The construction applies the original ƒ to many small pieces of the input to ƒ^* just to get one “hard-core” bit. The security of this bit may be smaller than any constant positive power of the security of ƒ. In fact, for inputs (to ƒ^*) of practical size, the pieces effected by ƒ are so small that ƒ can be inverted (and the “hard-core” bit computed) by exhaustive search.

In this paper we show that every one-way function, padded to the form ƒ(p, x) = (p, g(x)), ‖‖p‖‖ = ‖x‖, has by itself a hard-core predicate of the same (within a polynomial) security. Namely, we prove a conjecture of [Levin 87, sec. 5.6.2] that the scalar product of Boolean vectors p, x is a hard-core of every one-way function ƒ(p, x) = (p, g(x)). The result extends to multiple (up to the logarithm of security) such bits and to any distribution on the x's for which ƒ is hard to invert.

References

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A hard-core predicate for all one-way functions

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cover image ACM Conferences

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

February 1989

600 pages

ISBN:0897913078

DOI:10.1145/73007

Editor:
D. S. Johnson

Copyright © 1989 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 February 1989

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STOC89: 21st Annual ACM Symposium on the Theory of Computing

May 14 - 17, 1989

Washington, Seattle, USA

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STOC '89 Paper Acceptance Rate 56 of 196 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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https://doi.org/10.62056/a66c0l5vt
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