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.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Goldreich, O.: Foundation of Cryptography – Class Notes. Computer Science Department, Technion, Haifa, Israel (Spring 1989)
Goldreich, O.: Foundation of Cryptography – Fragments of a Book, Available from ECCC (February 1995)
Goldreich, O.: Foundation of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)
Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press, Cambridge (2008)
Goldreich, O., Levin, L.A.: A Hard-Core Predicate for all One-Way Functions. In: 21st STOC, pp. 25–32 (1989)
Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A Pseudorandom Generator from any One-way Function. SICOMP 28(4), 1364–1396 (1999); Combines papers of Impagliazzo et al. (21st STOC, 1989) and Håstad (22nd STOC, 1990)
Impagliazzo, R.: See [8], which appeared after our first posting (1994) (manuscript)
Impagliazzo, R.: Hard-core Distributions for Somewhat Hard Problems. In: 36th FOCS, pp. 538–545 (1995); This is a later version of [7]
Impagliazzo, R., Jaiswal, R., Kabanets, V.: Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification. In: 47th FOCS, pp. 187–196 (2006)
Impagliazzo, R., Jaiswal, R., Kabanets, V., Wigderson, A.: Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized. SIAM J. Comput. 39(4), 1637–1665 (2010); Preliminary version in 40th STOC (2008)
Impagliazzo, R., Wigderson, A.: P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma. In: 29th STOC, pp. 220–229 (1997)
Levin, L.A.: One-Way Functions and Pseudorandom Generators. Combinatorica 7(4), 357–363 (1987)
Levin, L.A.: Average Case Complete Problems. SICOMP 15, 285–286 (1986)
Nisan, N., Rudich, S., Saks, M.: Products and Help Bits in Decision Trees. In: 35th FOCS, pp. 318–329 (1994)
Nisan, N., Wigderson, A.: Hardness vs Randomness. JCSS 49(2), 149–167 (1994)
Viola, E., Wigderson, A.: Norms, XOR Lemmas, and Lower Bounds for Polynomials and Protocols. Theory of Computing 4(1), 137–168 (2008); Preliminary version in IEEE Conf. on Comput. Complex. (2007)
Yao, A.C.: Theory and Application of Trapdoor Functions. In: 23rd FOCS, pp. 80–91 (1982)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Goldreich, O., Nisan, N., Wigderson, A. (2011). On Yao’s XOR-Lemma. In: Goldreich, O. (eds) Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation. Lecture Notes in Computer Science, vol 6650. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22670-0_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-22670-0_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22669-4
Online ISBN: 978-3-642-22670-0
eBook Packages: Computer ScienceComputer Science (R0)