Abstract
We study the task of randomness extraction from sources that are distributed uniformly on an unknown algebraic variety. In other words, we are interested in constructing a function (an extractor) whose output is close to uniform even if the input is drawn uniformly from the set of solutions of an unknown system of low degree polynomials. This problem generalizes the problem of extraction from affine sources which has drawn a considerable amount of interest lately.
We present two constructions of explicit extractors for varieties. The first works for varieties of any size (including one-dimensional varieties or curves) and requires field size that is exponential in the overall dimension of the space. Our second extractor allows the field size to be polynomial in the degree of the equations defining the variety, but works only for varieties whose size is at least the square root of the total size of the space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Barak, A. Rao, R. Shaltiel & A. Wigderson (2006). 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, 671–680. ACM Press, New York, NY, USA. ISBN 1-59593-134-1.
Bombieri E. (1966) On Exponential Sums in Finite Fields. American Journal of Mathematics 88: 71–105
Bourgain J. (2007) On the construction of affine extractors. Geometric And Functional Analysis 17(1): 33–57
O. Chevassut, P. Fouque, P. Gaudry & D. Pointcheval (2006). The Twist-AUgmented Technique for Key Exchange. In Public Key Cryptography, Moti Yung, Yevgeniy Dodis, Aggelos Kiayias & Tal Malkin, editors, volume 3958 of Lecture Notes in Computer Science, 410–426. Springer. ISBN 3-540-33851-9. http://dblp.uni-trier.de/db/conf/pkc/pkc2006.html#ChevassutFGP06.
B. Chor, O. Goldreich, J. Håstad, J. Friedman, S. Rudich & R. Smolensky (1985). The Bit Extraction Problem of t-Resilient Functions (Preliminary Version). In Proceedings of the 26th Annual Symposium on Foundations of Computer Science, 396–407.
D. Cox, J. Little & D. O’Shea (2007). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics). Springer-Verlag New York, Inc., Secaucus, NJ, USA. ISBN 0387356509.
V. Danilov (1994). Algebraic varieties and schemes. In I. Shafarevich (Ed.), Algebraic Geometry I, Encyclopedia of Mathematical Sciences, Vol. 23, 167–297. Springer, Berlin.
Deligne P. (1974) La conjecture de Weil. I , Inst. Hautes Etudes Sci. Publ. Math. 43: 273–307
Z. Dvir, A. Gabizon & A. Wigderson (2009). Extractors And Rank Extractors For Polynomial Sources. Comput. Complex. 18, 1–58. ISSN 1016-3328. http://portal.acm.org/citation.cfm?id=1536240.1536241.
A. Gabizon & R. Raz (2008). Deterministic extractors for affine sources over large fields. Combinatorica 28, 415–440. ISSN 0209-9683. http://portal.acm.org/citation.cfm?id=1459886.1459887.
A. Gabizon, R. Raz & R. Shaltiel (2006). Deterministic Extractors for Bit-Fixing Sources by Obtaining an Independent Seed. SIAM J. Comput. 36, 1072–1094. ISSN 0097-5397. http://portal.acm.org/citation.cfm?id=1328008.1328010.
O. Goldreich (1995). Three XOR-Lemmas - An Exposition. Electronic Colloquium on Computational Complexity (ECCC) 2(056). http://citeseer.ist.psu.edu/goldreich95three.html.
N. Gurel (2005). Extracting bits from coordinates of a point of an elliptic curve. Technical report. Cryptology ePrint Archive, Report 2005/324, http://eprint.iacr.org/.
J. Harris (1992). Algebraic Geometry - A First Course. Springer.
R. Hartshorne (1977). Algebraic Geometry. Number 52 in Graduate Texts in Mathematics. Springer.
J. Kamp & D. Zuckerman (2006). Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography. SIAM J. Comput. 36, 1231–1247. ISSN 0097-5397. http://dx.doi.org/10.1137/S0097539705446846.
J. Kamp, A. Rao, S. Vadhan & D. Zuckerman (2011). Deterministic extractors for small-space sources. J. Comput. Syst. Sci. 77, 191–220. ISSN 0022-0000. http://dx.doi.org/10.1016/j.jcss.2010.06.014.
MorenoO. Kumar P. (1993) Minimum distance bounds for cyclic codes and Deligne’s theorem. IEEE Transactions on Information Theory 39(5): 1524–1534
A. Rao (2007). An Exposition of Bourgain’s 2-Source Extractor. Technical Report TR07-034, ECCC.
W. M. Schmidt (1976). Equations over Finite Fields: An Elementary Approach, volume 536. Springer-Verlag, Lecture Notes in Mathematics.
J. T. Schwartz (1980). Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. ACM 27(4), 701–717. ISSN 0004-5411.
I. R. Shafarevich (1994). Basic algebraic geometry. Springer-Verlag New York, Inc., New York, NY, USA. ISBN 0-387-54812-2.
R. Shaltiel (2008). How to get more mileage from randomness extractors. Random Struct. Algorithms 33, 157–186. ISSN 1042-9832. http://portal.acm.org/citation.cfm?id=1400123.1400124.
M. Sombra (1997). Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz. Journal of Pure and Applied Algebra 117-118, 565–599. ISSN 0022-4049. http://www.sciencedirect.com/science/article/B6V0K-3SP2C65-1P/2/6d4a177ce4576c10ad7dd8db8c0213d8.
L. Trevisan & S. Vadhan (2000). Extracting randomness from samplable distributions. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 32. IEEE Computer Society, Washington, DC, USA. ISBN 0-7695-0850-2.
Wooley T. (1996) A Note on Simultaneous Congruences. J. Number Theory 58: 288–297
R. Zippel (1979). Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, 216–226. Springer-Verlag. ISBN 3-540-09519-5.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dvir, Z. Extractors for varieties. comput. complex. 21, 515–572 (2012). https://doi.org/10.1007/s00037-011-0023-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00037-011-0023-3