Abstract
Condensers are functions which receive two inputs—a random string of bits chosen according to some unknown distribution and an independent uniform (short) seed—and output a string of bits which somehow preserves the randomness of the input. The parameters of interest here are the seed length, output length and how much randomness is preserved.
Here we present explicit algorithms for condensers which have constant seed size. Our constructions improve on previous constant-seed condensers of Barak et al (2005). When the input distribution has high min-entropy, we provide a condenser having optimal rate and seed chosen from {1, 2, 3}. The analysis of this construction is considerably simpler than those of previous constructions. For the low min-entropy regime, we provide a different construction which can be viewed as a pseudorandom coloring of hypergraphs. The analysis of this condenser involves a generalization of the celebrated Balog–Szemerédi–Gowers Theorem. As an example of the simplicity of the ideas behind this generalization, we improve Bourgain–Katz–Tao sum-product estimates in just a few lines.
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
Ta-Shma, A., Umans, C., Zuckerman, D.: Loss-less condensers, unbalanced expanders, and extractors. In: STOC: ACM Symposium on Theory of Computing (STOC) (2001)
Raz, R.: Extractors with weak random seeds. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 4(099) (2004)
Barak, B., Kindler, G., Shaltiel, R., Sudakov, B., Wigderson, A.: Simulating independence: new constructions of condensers, Ramsey graphs, dispersers, and extractors. In: STOC 2005: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pp. 1–10. ACM Press, New York (2005)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Electronic Colloquium on Computational Complexity (ECCC) (2005)
Reingold, O., Shaltiel, R., Wigderson, A.: Extracting randomness via repeated condensing. SIAM J. Comput. 35(5), 1185–1209 (2006)
Ta-Shma, A., Umans, C.: Better lossless condensers through derandomized curve samplers (2006)
Guruswami, V., Umans, C., Vadhan, S.: Extractors and condensers from univariate polynomials. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 6(134) (2006)
Bourgain, J., Katz, N., Tao, T.: A sum-product estimate in finite fields, and applications. Geometric And Functional Analysis 14, 27–57 (2004)
Hart, D., Iosevich, A., Solymosi, J.: Sum-product estimates in finite fields (2006)
Vu, V.: Sum-product estimates via directed expanders (2007)
Zuckerman: General weak random sources. In: FOCS: IEEE Symposium on Foundations of Computer Science (FOCS) (1990)
Barak, B., Impagliazzo, R., Wigderson, A.: Extracting randomness using few independent sources. In: FOCS, pp. 384–393 (2004)
Ruzsa, I.: An analog of Freiman’s theorem in groups, Structure theory of set adition. Astérisque 258, 323–326 (1999)
Nathanson, M.B.: Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics, vol. 165. Springer, New York (1996)
Gowers, W.T.: A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8(3), 529–551 (1998)
Sudakov, B., Szemerédi, E., Vu, V.H.: On a question of Erdős and Moser. Duke Math. J. 129, 129–155 (2005)
Konyagin, S.V.: A sum-product estimate in fields of prime order (2003)
Dvir, Z., Raz, R.: Analyzing linear mergers. In: Electronic Colloquium on Computational Complexity (ECCC) (2005)
Tao, T., Vu, V.H.: Additive Combinatorics. In: Cambridge Studies in Advanced Mathematics (2006)
Garaev, M.Z.: An explicit sum-product estimate in \(\mathbb{F}_p\). In: ArXiv Mathematics e-prints (February 2007)
Hawk Katz, N., Shen, C.Y.: Garaev’s Inequality in finite fields not of prime order. In: ArXiv Mathematics e-prints (March 2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dellamonica, D. (2008). Simpler Constant-Seed Condensers. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_57
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_57
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)