Abstract
We provide an exposition of three lemmas that relate general properties of distributions over bit strings to the exclusive-or (xor) of values of certain bit locations.
The first XOR-Lemma, commonly attributed to Umesh Vazirani (1986), relates the statistical distance of a distribution from the uniform distribution over bit strings to the maximum bias of the xor of certain bit positions. The second XOR-Lemma, due to Umesh and Vijay Vazirani (19th STOC, 1987), is a computational analogue of the first. It relates the pseudorandomness of a distribution to the difficulty of predicting the xor of bits in particular or random positions. The third Lemma, due to Goldreich and Levin (21st STOC, 1989), relates the difficulty of retrieving a string and the unpredictability of the xor of random bit positions. The most notable XOR Lemma – that is the so-called Yao XOR Lemma – is not discussed here.
We focus on the proofs of the aforementioned three lemma. Our exposition deviates from the original proofs, yielding proofs that are believed to be simpler, of wider applicability, and establishing somewhat stronger quantitative results. Credits for these improved proofs are due to several researchers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alexi, W., Chor, B., Goldreich, O., Schnorr, C.P.: RSA and Rabin Functions: Certain Parts Are As Hard As the Whole. SIAM Journ. on Computing 1988, 194–209 (1984)
Alon, N., Goldreich, O., Håstad, J., Peralta, R.: Simple Constructions of Almost k-wise Independent Random Variables. Journal of Random Structures and Algorithms 3(3), 289–304 (1992)
Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols and logspace-hard pseudorandom sequences. In: 21st STOC, pp. 1–11 (1989)
Blum, M., Micali, S.: How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. SIAM Journ. on Computing 1984, 850–864 (1982); Preliminary version in 23rd FOCS 1982
Chor, B., Friedmann, J., Goldreich, O., Hastad, J., Rudich, S., Smolansky, R.: The Bit Extraction Problem or t-Resilient Functions. In: Proc. of the 26th IEEE Symp. on Foundation Of Computer Science (FOCS), pp. 396–407 (1985)
Erdos, P., Spenser, J.: Probabilistic Methods in Combinatorics. Academic Press, New York (1974)
Goldreich, O.: Foundation of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)
Goldreich, O.: Foundation of Cryptography: Basic Applications. Cambridge University Press, Cambridge (2004)
Goldreich, O., Goldwasser, S., Micali, S.: How to Construct Random Functions. Jour. of the ACM 33(4), 792–807 (1986)
Goldreich, O., Levin, L.A.: Hard-core Predicates for any One-Way Function. In: 21st STOC, pp. 25–32 (1989)
Goldreich, O., Nisan, N., Wigderson, A.: On Yao’s XOR-Lemma. In: Goldreich, O., et al.: Studies in Complexity and Cryptography. LNCS, vol. 6650, pp. 273–301. Springer, Heidelberg (2011)
Goldreich, O., Rubinfeld, R., Sudan, M.: Learning polynomials with queries: the highly noisy case. SIAM J. Discrete Math. 13(4), 535–570 (2000)
Goldwasser, S., Micali, S.: Probabilistic Encryption. JCSS 28(2), 270–299 (1982); Preliminary version in 14th STOC 1982
Kaliski Jr., B.S.: Elliptic Curves and Cryptography: A Pseudorandom Bit Generator and Other Tools, Ph.D. Thesis, LCS, MIT (1988)
Levin, L.A.: One-Way Function and Pseudorandom Generators. Combinatorica 7(4), 357–363 (1987); A preliminary version in 19th STOC 1985
Naor, J., Naor, M.: Small-bias Probability Spaces: Efficient Constructions and Applications. In: 22nd STOC, pp. 213–223 (1990)
Nisan, N.: Pseudorandom Generators for Space-Bounded Computations. In: 22nd STOC, pp. 204–212 (1990)
Rabin, M.O.: Digitalized Signatures and Public Key Functions as Intractable as Factoring, MIT/LCS/TR-212 (1979)
Rivest, R., Shamir, A., Adleman, L.: A Method for Obtaining Digital Signatures and Public Key Cryptosystems. Comm. ACM 21, 120–126 (1978)
Vazirani, U.V.: Randomness, Adversaries and Computation, Ph.D. Thesis, EECS, UC Berkeley (1986)
Vazirani, U.V.: Efficiency Considerations in Using Semi-random Sources. In: Proc. 19th ACM Symp. on Theory of Computing, pp. 160–168 (1987)
Vazirani, U.V., Vazirani, V.V.: Efficient and Secure Pseudo-Random Number Generation. In: Proc. 25th IEEE Symp. on Foundation of Computer Science, pp. 458–463 (1984)
Yao, A.C.: Theory and Applications of Trapdoor Functions. In: Proc. of the 23rd IEEE Symp. on Foundation of Computer Science, 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. (2011). Three XOR-Lemmas — An Exposition. 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_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-22670-0_22
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)