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Extractors and Lower Bounds for Locally Samplable Sources

Published:01 March 2012Publication History
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Abstract

We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number d of the random input bits. As our main result, we construct a deterministic extractor that, given any d-local source with min-entropy k on n bits, extracts Ω(k2/nd) bits that are 2−nΩ(1)-close to uniform, provided do(log n) and kn2/3+γ (for arbitrarily small constants γ > 0).

Using our result, we also improve a result of Viola [2010] who proved a 1/2 − O(1/log n) statistical distance lower bound for o(log n)-local samplers trying to sample input-output pairs of an explicit boolean function, assuming the samplers use at most n + n1−δ random bits for some constant δ > 0. Using a different function, we simultaneously improve the lower bound to 1/2 − 2−nΩ(1) and eliminate the restriction on the number of random bits.

References

  1. Applebaum, B. 2011. Pseudorandom generators with long stretch and low locality from random local one-way functions. Tech. rep. TR11-007, Electronic Colloquium on Computational Complexity.Google ScholarGoogle Scholar
  2. Applebaum, B., Ishai, Y., and Kushilevitz, E. 2006. Cryptography in NC0. SIAM J. Comput. 36, 4, 845--888. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Applebaum, B., Ishai, Y., and Kushilevitz, E. 2008. On pseudorandom generators with linear stretch in NC0. Comput. Complex. 17, 1, 38--69. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Applebaum, B., Bogdanov, A., and Rosen, A. 2011. A dichotomy for local small-bias generators. Tech. rep. TR11-126, Electronic Colloquium on Computational Complexity.Google ScholarGoogle Scholar
  5. Arora, S., Steurer, D., and Wigderson, A. 2009. Towards a study of low-complexity graphs. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming. 119--131. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Barak, B., Impagliazzo, R., and Wigderson, A. 2006a. Extracting randomness using few independent sources. SIAM J. Comput. 36, 4, 1095--1118. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Barak, B., Rao, A., Shaltiel, R., and Wigderson, A. 2006b. 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. In Proceedings of the 38th ACM Symposium on Theory of Computing. 671--680. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Barak, B., Kindler, G., Shaltiel, R., Sudakov, B., and Wigderson, A. 2010. Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors. J. ACM 57, 4. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Bellare, M. and Rompel, J. 1994. Randomness-efficient oblivious sampling. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science. 276--287. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Ben-Sasson, E. and Gabizon, A. 2011. Extractors for polynomials sources over constant-size fields of small characteristic. Tech. rep. TR11-129, Electronic Colloquium on Computational Complexity.Google ScholarGoogle Scholar
  11. Bogdanov, A. and Qiao, Y. 2009. On the security of Goldreich’s one-way function. In Proceedings of the 13th International Workshop on Randomization and Computation. 392--405. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Bourgain, J. 2005. More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1, 1--32.Google ScholarGoogle ScholarCross RefCross Ref
  13. Bourgain, J. 2007. On the construction of affine-source extractors. Geom. Funct. Anal. 1, 33--57.Google ScholarGoogle ScholarCross RefCross Ref
  14. Chor, B. and Goldreich, O. 1988. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 2, 230--261. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Chor, B., Friedman, J., Goldreich, O., Håstad, J., Rudich, S., and Smolensky, R. 1985. The bit extraction problem or t-resilient functions. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science. 396--407. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Cook, J., Etesami, O., Miller, R., and Trevisan, L. 2009. Goldreich’s one-way function candidate and myopic backtracking algorithms. In Proceedings of the 6th Theory of Cryptography Conference. 521--538. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Cryan, M. and Miltersen, P. B. 2001. On pseudorandom generators in NC0. In Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science. 272--284. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. De, A. and Trevisan, L. 2009. Extractors using hardness amplification. In Proceedings of the 13th International Workshop on Randomization and Computation. 462--475. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. DeVos, M. and Gabizon, A. 2010. Simple affine extractors using dimension expansion. In Proceedings of the 25th IEEE Conference on Computational Complexity. 50--57. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Dodis, Y., Elbaz, A., Oliveira, R., and Raz, R. 2004. Improved randomness extraction from two independent sources. In Proceedings of the 8th International Workshop on Randomization and Computation. 334--344.Google ScholarGoogle Scholar
  21. Dvir, Z. 2009. Extractors for varieties. In Proceedings of the 24th IEEE Conference on Computational Complexity. 102--113. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Dvir, Z., Gabizon, A., and Wigderson, A. 2009. Extractors and rank extractors for polynomial sources. Comput. Complex. 18, 1, 1--58. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Dziembowski, S. and Maurer, U. 2004. Optimal randomizer efficiency in the bounded-storage model. J. Cryptol. 17, 1, 5--26. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Gabizon, A. and Raz, R. 2008. Deterministic extractors for affine sources over large fields. Combinatorica 28, 4, 415--440. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Gabizon, A., Raz, R., and Shaltiel, R. 2006. Deterministic extractors for bit-fixing sources by obtaining an independent seed. SIAM J. Comput. 36, 4, 1072--1094. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Goldreich, O. 2011. Candidate one-way functions based on expander graphs. Studies Complex. Cryptograph., 76--87.Google ScholarGoogle Scholar
  27. Goldwasser, S., Gutfreund, D., Healy, A., Kaufman, T., and Rothblum, G. 2007. Verifying and decoding in constant depth. In Proceedings of the 39th ACM Symposium on Theory of Computing. 440--449. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Guruswami, V. and Rudra, A. 2008. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Trans. Inf. Theory 54, 1, 135--150. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Guruswami, V., Umans, C., and Vadhan, S. 2009. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. J. ACM 56, 4. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. Håstad, J. 1986. Almost optimal lower bounds for small depth circuits. In Proceedings of the 18th ACM Symposium on Theory of Computing. 6--20. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Håstad, J. 1987. One-way permutations in NC0. Inf. Process. Lett. 26, 3, 153--155. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Ishai, Y., Kushilevitz, E., Ostrovsky, R., and Sahai, A. 2008. Cryptography with constant computational overhead. In Proceedings of the 40th ACM Symposium on Theory of Computing. 433--442. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Kamp, J. and Zuckerman, D. 2007. Deterministic extractors for bit-fixing sources and exposure-resilient cryptography. SIAM J. Comput. 36, 5, 1231--1247. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Kamp, J., Rao, A., Vadhan, S., and Zuckerman, D. 2011. Deterministic extractors for small-space sources. J. Comput. Syst. Sci. 77, 1, 191--220. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. Li, X. 2011a. Improved constructions of three source extractors. In Proceedings of the 26th IEEE Conference on Computational Complexity. 126--136. Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. Li, X. 2011b. A new approach to affine extractors and dispersers. In Proceedings of the 26th IEEE Conference on Computational Complexity. 137--147. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Lovett, S. and Viola, E. 2011. Bounded-depth circuits cannot sample good codes. In Proceedings of the 26th IEEE Conference on Computational Complexity. 243--251. Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. Lu, C.-J. 2004. Encryption against storage-bounded adversaries from on-line strong extractors. J. Cryptol. 17, 1, 27--42. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Mossel, E., Shpilka, A., and Trevisan, L. 2006. On epsilon-biased generators in NC0. Random Struct. Algor. 29, 1, 56--81. Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. Nisan, N. and Zuckerman, D. 1996. Randomness is linear in space. J. Comput. Syst. Sci. 52, 1, 43--52. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Rao, A. 2008. A 2-source almost-extractor for linear entropy. In Proceedings of the 12th International Workshop on Randomization and Computation. 549--556. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. Rao, A. 2009a. Extractors for a constant number of polynomially small min-entropy independent sources. SIAM J. Comput. 39, 1, 168--194. Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Rao, A. 2009b. Extractors for low-weight affine sources. In Proceedings of the 24th IEEE Conference on Computational Complexity. 95--101. Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Rao, A. and Zuckerman, D. 2008. Extractors for three uneven-length sources. In Proceedings of the 12th International Workshop on Randomization and Computation. 557--570. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Raz, R. 2005. Extractors with weak random seeds. In Proceedings of the 37th ACM Symposium on Theory of Computing. 11--20. Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. Raz, R. and Yehudayoff, A. 2011. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. J. Comput. Syst. Sci. 77, 1, 167--190. Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Raz, R., Reingold, O., and Vadhan, S. 1999. Error reduction for extractors. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science. 191--201. Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Raz, R., Reingold, O., and Vadhan, S. 2002. Extracting all the randomness and reducing the error in Trevisan’s extractors. J. Comput. Syst. Sci. 65, 1, 97--128. Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Schmidt, J., Siegel, A., and Srinivasan, A. 1995. Chernoff-Hoeffding bounds for applications with limited independence. SIAM J. Discr. Math. 8, 2, 223--250. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Shaltiel, R. 2002. Recent developments in explicit constructions of extractors. Bull. Euro. Assoc. Theor. Comput. Sci. 77, 67--95.Google ScholarGoogle Scholar
  51. Shaltiel, R. 2008. How to get more mileage from randomness extractors. Random Struct. Algor. 33, 2, 157--186. Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Shaltiel, R. 2011. An introduction to randomness extractors. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming. 21--41. Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. Shoup, V. 1988. New algorithms for finding irreducible polynomials over finite fields. In Proceedings of the 29th IEEE Symposium on Foundations of Computer Science. 283--290. Google ScholarGoogle ScholarDigital LibraryDigital Library
  54. Sudan, M. 2002. Essential coding theory---Course notes. http://theory.lcs.mit.edu/~madhu/coding/.Google ScholarGoogle Scholar
  55. Tauman Kalai, Y., Li, X., and Rao, A. 2009. 2-source extractors under computational assumptions and cryptography with defective randomness. In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science. 617--626. Google ScholarGoogle ScholarDigital LibraryDigital Library
  56. Trevisan, L. 2001. Extractors and pseudorandom generators. J. ACM 48, 4, 860--879. Google ScholarGoogle ScholarDigital LibraryDigital Library
  57. Trevisan, L. and Vadhan, S. 2000. Extracting randomness from samplable distributions. In Proceedings of the 41st IEEE Symposium on Foundations of Computer Science. 32--42. Google ScholarGoogle ScholarDigital LibraryDigital Library
  58. Vadhan, S. 2004. Constructing locally computable extractors and cryptosystems in the bounded-storage model. J. Cryptol. 17, 1, 43--77. Google ScholarGoogle ScholarDigital LibraryDigital Library
  59. Vadhan, S. 2011. Pseudorandomness. In Foundations and Trends in Theoretical Computer Science. To appear. Google ScholarGoogle ScholarDigital LibraryDigital Library
  60. Viola, E. 2010. The complexity of distributions. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science. 202--211. Google ScholarGoogle ScholarDigital LibraryDigital Library
  61. Viola, E. 2011. Extractors for circuit sources. In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science. To appear. Google ScholarGoogle ScholarDigital LibraryDigital Library
  62. Yao, A. 1985. Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science. 1--10. Google ScholarGoogle ScholarDigital LibraryDigital Library
  63. Yehudayoff, A. 2011. Affine extractors over prime fields. Combinatorica 31, 2, 245--256. Google ScholarGoogle ScholarDigital LibraryDigital Library
  64. Zimand, M. 2010. Simple extractors via constructions of cryptographic pseudo-random generators. Theor. Comput. Sci. 411, 10, 1236--1250. Google ScholarGoogle ScholarDigital LibraryDigital Library

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 4, Issue 1
      March 2012
      76 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2141938
      Issue’s Table of Contents

      Copyright © 2012 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 March 2012
      • Accepted: 1 December 2011
      • Revised: 1 October 2011
      • Received: 1 July 2011
      Published in toct Volume 4, Issue 1

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