On Yao's XOR-Lemma

Oded Goldreich*, Noam Nisan, Avi Wigderson

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

38 Scopus citations


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 languageAmerican English
Title of host publicationStudies in Complexity and Cryptography
Subtitle of host publicationMiscellanea on the Interplay between Randomness and Computation
EditorsOded Goldreich
Number of pages29
StatePublished - 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6650 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


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


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