On Yao's XOR-Lemma

Oded Goldreich; Noam Nisan; Avi Wigderson

doi:10.1007/978-3-642-22670-0_23

On Yao's XOR-Lemma

Oded Goldreich, Noam Nisan, Avi Wigderson

Weizmann Institute of Science, The Hebrew University of Jerusalem

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

A fundamental lemma of Yao states that computational weak-unpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao's Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the original function. This statement turns out to be easier to prove than the XOR-Lemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XOR-Lemma.

Original language	English
Title of host publication	Studies in Complexity and Cryptography
Subtitle of host publication	Miscellanea on the Interplay between Randomness and Computation
Editors	Oded Goldreich
Chapter	23
Pages	273-301
Number of pages	29
DOIs	https://doi.org/10.1007/978-3-642-22670-0_23
State	Published - 2011

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6650 LNCS

Keywords

Direct Product Lemma
Hard-Core Predicates
Hard-Core Regions
One-Way Functions
Yao's XOR Lemma

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-642-22670-0_23

Cite this

Goldreich, O., Nisan, N., & Wigderson, A. (2011). On Yao's XOR-Lemma. In O. Goldreich (Ed.), Studies in Complexity and Cryptography: Miscellanea on the Interplay between Randomness and Computation (pp. 273-301). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6650 LNCS). https://doi.org/10.1007/978-3-642-22670-0_23

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abstract = "A fundamental lemma of Yao states that computational weak-unpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao's Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the original function. This statement turns out to be easier to prove than the XOR-Lemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XOR-Lemma.",

keywords = "Direct Product Lemma, Hard-Core Predicates, Hard-Core Regions, One-Way Functions, Yao's XOR Lemma",

author = "Oded Goldreich and Noam Nisan and Avi Wigderson",

note = "We wish to thank Mike Saks for useful discussions regarding Levin{\textquoteright}s proof of the XOR Lemma. We also thank Salil Vadhan and Ronen Shaltiel for pointing out errors in previous versions, and for suggesting ways to fix these errors.",

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doi = "10.1007/978-3-642-22670-0\_23",

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Goldreich, O , Nisan, N & Wigderson, A 2011, On Yao's XOR-Lemma. in O Goldreich (ed.), Studies in Complexity and Cryptography: Miscellanea on the Interplay between Randomness and Computation. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6650 LNCS, pp. 273-301. https://doi.org/10.1007/978-3-642-22670-0_23

On Yao's XOR-Lemma. / Goldreich, Oded ; Nisan, Noam; Wigderson, Avi.
Studies in Complexity and Cryptography: Miscellanea on the Interplay between Randomness and Computation. ed. / Oded Goldreich. 2011. p. 273-301 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6650 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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Goldreich O , Nisan N, Wigderson A. On Yao's XOR-Lemma. In Goldreich O, editor, Studies in Complexity and Cryptography: Miscellanea on the Interplay between Randomness and Computation. 2011. p. 273-301. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-22670-0_23

On Yao's XOR-Lemma

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