Extractors for a constant number of polynomially small min-entropy independent sources

SIAM Journal on Computing

Volumn 39, Issue 1, 2009, Pages 168-194

  Rao, Anup ^a  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

Extractor; Independent sources; Ramsey graphs

Indexed keywords

BLACK BOXES; COMBINATORICS; EXTRACTOR; INDEPENDENT SOURCES; LINEAR OUTPUT; MIN-ENTROPY; NEW CONSTRUCTION; NEW YORK; RAMSEY GRAPHS; RANDOM SEEDS; RANDOM SOURCES; SOURCE EXTRACTOR;

COMPUTATION THEORY; CONDENSERS (LIQUEFIERS); ENTROPY; MACHINE TOOLS; NUMBER THEORY;

GRAPH THEORY;

EID: 67650164537     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/060671218     Document Type: Conference Paper

References (33)

1. B. BARAK, R. IMPAGLIAZZO, AND A. WIGDERSON, Extracting randomness using few independent sources, SIAM J. Comput., 36 (2006), pp. 1095-1118.
2. B. BARAK, G. KINDLER, R. SHALTIEL, B. SUDAKOV, AND A. WIGDERSON, Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 1-10.
3. B. BARAK, A. RAO, R. SHALTIEL, AND A. WIGDERSON, 2 source dispersers for subpolynomial entropy and Ramsey graphs beating the Frankl-Wilson construction, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 671-680.
4. M. BLUM, Independent unbiased coin flips from a correlated biased source - A finite state Markov chain, Combinatorica, 6 (1986), pp. 97-108.
5. J. BOURGAIN, More on the sum-product phenomenon in prime fields and its applications, Int. J. Number Theory, 1 (2005), pp. 1-32.
6. J. BOURGAIN, N. KATZ, AND T. TAO, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal., 14 (2004), pp. 27-57.
7. M. CAPALBO, O. REINGOLD, S. VADHAN, AND A. WIGDERSON, Randomness conductors and constant-degree lossless expanders, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 659-668.
8. B. CHOR, J. FRIEDMAN, O. GOLDREICH, J. HÅSTAD, S. RUDICH, AND R. SMOLENSKY, The bit extraction problem or t-resilient functions, in Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, 1985, pp. 396-407.
9. B. CHOR AND O. GOLDREICH, Unbiased bits from sources of weak randomness and probabilistic communication complexity, SIAM J. Comput., 17 (1988), pp. 230-261.
10. K.-M. CHUNG AND S. VADHAN, Personal communication.
11. A. COHEN AND A. WIGDERSON, Dispersers, deterministic amplification, and weak random sources, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, 1989, pp. 14-19.
12. A. GABIZON AND R. RAZ, Deterministic extractors for affine sources over large fields, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 407-418.
13. A. GABIZON, R. RAZ, AND R. SHALTIEL, Deterministic extractors for bit-fixing sources by obtaining an independent seed, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 394-403.
14. V. GURUSWAMI, C. UMANS, AND S. VADHAN, Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes, in Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, 2007, pp. 96-108.
15. J. KAMP AND D. ZUCKERMAN, Deterministic Extractors for Affine Sources from Bent Functions, manuscript.
16. S. KONYAGIN, A Sum-Product Estimate in Fields of Prime Order, Technical report, ArXiv, 2003, available online at http://arxiv.org/abs/math.NT/0304217.
17. C.-J. Lu, O. REINGOLD, S. VADHAN, AND A. WIGDERSON, Extractors: Optimal up to constant factors, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 602-611.
18. E. MOSSEL AND C. UMANS, On the complexity of approximating the VC dimension, J. Comput. System Sci., 65 (2002), pp. 660-671.
19. N. NISAN, Extracting randomness: How and why - A survey, in Proceedings of the 11th Annual IEEE Conference on Computational Complexity, 1996, pp. 44-58.
20. N. NISAN AND A. TA-SHMA, Extracting randomness: A survey and new constructions, J. Comput. System Sci., 58 (1999), pp. 148-173.
21. N. NISAN AND D. ZUCKERMAN, More deterministic simulation in logspace, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993, pp. 235-244.
22. R. RAZ, Extractors with weak random seeds, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 11-20.
23. R. RAZ, O. REINGOLD, AND S. VADHAN, Extracting all the randomness and reducing the error in Trevisan's extractors, J. Comput. System Sci., 65 (2002), pp. 97-128.
24. O. REINGOLD, R. SHALTIEL, AND A. WIGDERSON, Extracting randomness via repeated condensing, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000, pp. 22-31.
25. M. SANTHA AND U. V. VAZIRANI, Generating quasirandom sequences from semirandom sources, J. Comput. System Sci., 33 (1986), pp. 75-87.
26. R. SHALTIEL, Recent developments in explicit constructions of extractors, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 77 (2002), pp. 67-95.
27. R. SHALTIEL, HOW to get more mileage from randomness extractors, Random Structures Algorithms, 33 (2008), pp. 157-186.
28. A. TA-SHMA, Refining randomness, in ECCCTH: Electronic Colloquium on Computational Complexity, theses, 1996.
29. A. TA-SHMA, C. UMANS, AND D. ZUCKERMAN, Loss-less condensers, unbalanced expanders, and extractors, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 143-152.
30. L. TREVISAN, Extractors and pseudorandom generators, J. ACM, 48 (2001), pp. 860-879.
31. L. TREVISAN AND S. VADHAN, Extracting randomness from samplable distributions, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, pp. 32-42.
32. U. VAZIRANI, Towards a strong communication complexity theory or generating quasi-random sequences from two communicating slightly-random sources (extended abstract), in Proceedings of the 17th Annual ACM Symposium on Theory of Computing, ACM, New York, 1985, pp. 366-378.
33. J. VON NEUMANN, Various techniques used in connection with random digits, Applied Math Series, 12 (1951), pp. 36-38.