Extracting randomness via repeated condensing

SIAM Journal on Computing

Volumn 35, Issue 5, 2006, Pages 1185-1209

  Reingold, Omer ^a   Shaltiel, Ronen ^b   Wigderson, Avi ^c  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b UNIVERSITY OF HAIFA   (Israel)

^c INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

Derandomization; Randomness condensers; Randomness extractors

Indexed keywords

BLOCK EXTRACTION; EXTRACTORS; POLYLOGARITHMIC LOSS; RANDOM BITS;

ENTROPY; HARDNESS; OPTIMAL CONTROL SYSTEMS; PARAMETER ESTIMATION; POLYNOMIALS; TECHNICAL PRESENTATIONS;

RANDOM PROCESSES;

EID: 33750154710     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539703431032     Document Type: Article

References (32)

1. M. BLUM, Independent unbiased coin flips from a correlated biased source - a finite state Markov chain, Combinatorica, 6 (1986), pp. 97-108.
2. M. R. CAPALBO, O. REINGOLD, S. P. 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.
3. B. CHOR AND O. GOLDREICH, On the power of two-point based sampling, J. Complexity, 5 (1989), pp. 96-106.
4. O. GOLDREICH AND A. WIGDERSON, Tiny families of functions with random properties: A quality-size trade-off for hashing, Random Structures Algorithms, 11 (1997), pp. 315-343.
5. R. IMPAGLIAZZO, private communication, 1999.
6. R. IMPAGLIAZZO, R. SHALTIEL, AND A. WIGDERSON, Near-optimal conversion of hardness into pseudo-randomness, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1999, pp. 181-190.
7. R. IMPAGLIAZZO, R. SHALTIEL, AND A. WIGDERSON, Extractors and pseudo-random generators with optimal seed length, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (Portland, OR), ACM, New York, 2000, pp. 1-10.
8. J. JUSTESEN, A class of constructive asymptotically good algebraic codes, IEEE Trans. Inform. Theory, 18 (1972), pp. 652-656.
9. 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.
10. J. NAOR AND M. NAOR, Small-bias probability spaces: Efficient constructions and applications, SIAM J. Comput., 22 (1993), pp. 838-856.
11. N. NISAN, Extracting randomness: How and why: A survey, in Proceedings of the Eleventh Annual IEEE Conference on Computational Complexity (Philadelphia, PA), IEEE, Los Alamitos, CA, 1996, pp. 44-58.
12. N. NISAN AND A. TA-SHMA, Extracting randomness: A survey and new constructions, J. Comput. System Sci., 58 (1999), pp. 148-173.
13. N. NISAN AND D. ZUCKERMAN, Randomness is linear in space, J. Comput. System Sci., 52 (1996), pp. 43-52.
14. J. RADHAKRISHNAN AND A. TA-SHMA, Bounds for dispersera, extractors, and depth-two superconcentrators, SIAM J. Discrete Math., 13 (2000), pp. 2-24.
15. R. RAZ AND O. REINGOLD, On recycling the randomness of states in space bounded computation, in Proceedings of the ACM Symposium on Theory of Computing, ACM, New York, 1999, pp. 159-168.
16. R. RAZ, O. REINGOLD, AND S. VADHAN, Error reduction for extractors, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1999, pp. 191-201.
17. R. RAZ, O. REINGOLD, AND S. VADHAN, Extracting and the randomness and reducing the error in Trevisan's extractors, J. Comput. System Sci., 65 (2002), pp. 97-128.
18. 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, IEEE, Los Alamitos, CA, 2000, pp. 22-31.
19. O. REINGOLD, S. VADHAN, AND A. WIGDERSON, Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA), IEEE, Los Alamitos, CA, 2000, pp. 3-13.
20. M. SANTHA AND U. V. VAZIRANI, Generating quasi-random sequences from semi-random sources, J. Comput. System Sci., 33 (1986), pp. 75-87.
21. R. SHALTIEL, Recent developments in extractors, Bull. Eur. Assoc. Theor. Comput. Sci., 77 (2002), pp. 67-95.
22. R. SHALTIEL AND C. UMANS, Simple extractors for all min-entropies and a new pseudo-random generator, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (Las Vegas, NV), 2001, IEEE, Piscataway, NJ, pp. 648-657.
23. A. SRINIVASAN AND D. ZUCKERMAN, Computing with very weak random sources, SLAM J. Comput., 28 (1999), pp. 1433-1459.
24. A. TA-SHMA, On extracting randomness from weak random sources (extended abstract), in Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, (Philadelphia, PA), ACM, New York, 1996, pp. 276-285.
25. 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 (Hersonissos, Crete, Greece), ACM, New York, 2001, pp. 143-152.
26. A. TA-SHMA, D. ZUCKERMAN, AND S. SAFRA, Extractors from Reed-Muller codes, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2001, pp. 181-190.
27. L. TREVISAN, Extractors and pseudorandom generators, J. ACM, 48 (2001), pp. 860-879.
28. J. VON NEUMANN, Various techniques used in connection with random digits, Nat. Bur. Standards Appl. Math. Ser., 12 (1951), pp. 36-38.
29. A. WIGDERSON AND D. ZUCKERMAN, Expanders that beat the eigenvalue bound: Explicit construction and applications, in Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (San Diego, CA), ACM, New York, 1993, pp. 245-251.
30. D. ZUCKERMAN, General weak random sources, in Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science (St. Louis, MO, IEEE), vol. II, IEEE, Piscataway, NJ, 1990, pp. 534-543.
31. D. ZUCKERMAN, Simulating BPP using a general weak random source, Algorithmica, 16 (1996), pp. 367-391.
32. D. ZUCKERMAN, Randomness-optimal oblivious sampling, Random Structures Algorithms, 11 (1997), pp. 345-367.