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Constant-Error Pseudorandomness Proofs from Hardness Require Majority

Published:17 June 2019Publication History
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Abstract

Research in the 1980s and 1990s showed how to construct a pseudorandom generator from a function that is hard to compute on more than 99% of the inputs. A more recent line of works showed, however, that if the generator has small error, then the proof of correctness cannot be implemented in subclasses of TC0, and hence the construction cannot be applied to the known hardness results. This article considers a typical class of pseudorandom generator constructions, and proves an analogous result for the case of large error.

References

  1. Sergei Artemenko and Ronen Shaltiel. 2014. Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification. Computational Complexity 23, 1 (2014), 43--83. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. James Aspnes, Richard Beigel, Merrick Furst, and Steven Rudich. 1994. The expressive power of voting polynomials. Combinatorica. A Journal on Combinatorics and the Theory of Computing 14, 2 (1994), 135--148.Google ScholarGoogle Scholar
  3. Bill Fefferman, Ronen Shaltiel, Christopher Umans, and Emanuele Viola. 2013. On beating the hybrid argument. Theory of Computing 9 (2013), 809--843.Google ScholarGoogle ScholarCross RefCross Ref
  4. Oded Goldreich, Noam Nisan, and Avi Wigderson. 1995. On Yao’s XOR Lemma. Technical Report TR95--050. Electronic Colloquium on Computational Complexity. www.eccc.uni-trier.de/.Google ScholarGoogle Scholar
  5. Aryeh Grinberg, Ronen Shaltiel, and Emanuele Viola. 2018. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In IEEE Symposium on Foundations of Computer Science (FOCS’18). Available at http://www.ccs.neu.edu/home/viola/.Google ScholarGoogle ScholarCross RefCross Ref
  6. Dan Gutfreund and Guy Rothblum. 2008. The complexity of local list decoding. In 12th International Workshop on Randomization and Computation (RANDOM’08).Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Jeff Kinne, Dieter van Melkebeek, and Ronen Shaltiel. 2012. Pseudorandom generators, typically-correct derandomization, and circuit lower bounds. Computational Complexity 21, 1 (2012), 3--61. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Adam Klivans and Rocco A. Servedio. 2003. Boosting and hard-core sets. Machine Learning 53, 3 (2003), 217--238. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Adam R. Klivans. 2001. On the derandomization of constant depth circuits. In Workshop on Randomization and Computation (RANDOM’01). Springer.Google ScholarGoogle ScholarCross RefCross Ref
  10. Shachar Lovett and Emanuele Viola. 2012. Bounded-depth circuits cannot sample good codes. Computational Complexity 21, 2 (2012), 245--266. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Chi-Jen Lu, Shi-Chun Tsai, and Hsin-Lung Wu. 2011. Complexity of hard-core set proofs. Computational Complexity 20, 1 (2011), 145--171. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Moni Naor and Omer Reingold. 2004. Number-theoretic constructions of efficient pseudo-random functions. Journal of the ACM 51, 2 (2004), 231--262. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Noam Nisan. 1991. Pseudorandom bits for constant depth circuits. Combinatorica. A Journal on Combinatorics and the Theory of Computing 11, 1 (1991), 63--70.Google ScholarGoogle Scholar
  14. Noam Nisan and Avi Wigderson. 1994. Hardness vs randomness. Journal of Computer and System Sciences 49, 2 (1994), 149--167. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Alexander Razborov. 1987. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Akademiya Nauk SSSR. Matematicheskie Zametki 41, 4 (1987), 598--607. English translation in Mathematical Notes of the Academy of Sciences of the USSR, 41, 4 (1987), 333--338.Google ScholarGoogle Scholar
  16. Alexander Razborov and Steven Rudich. 1997. Natural proofs. Journal of Computer and System Sciences 55, 1 (Aug. 1997), 24--35.Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Ronen Shaltiel and Emanuele Viola. 2010. Hardness amplification proofs require majority. SIAM Journal on Computing 39, 7 (2010), 3122--3154.Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Roman Smolensky. 1987. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In 19th ACM Symposium on the Theory of Computing (STOC’87). ACM, 77--82.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Emanuele Viola. 2004. The complexity of constructing pseudorandom generators from hard functions. Computational Complexity 13, 3--4 (2004), 147--188. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Emanuele Viola. 2006. The Complexity of Hardness Amplification and Derandomization. Ph.D. Dissertation. Harvard University. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Emanuele Viola. 2007. Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates. SIAM Journal on Computing 36, 5 (2007), 1387--1403. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Emanuele Viola. 2009. On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science 5, 1 (2009), 1--72. Google ScholarGoogle ScholarDigital LibraryDigital Library

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 11, Issue 4
        December 2019
        252 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3331049
        Issue’s Table of Contents

        Copyright © 2019 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 17 June 2019
        • Accepted: 1 March 2019
        • Revised: 1 February 2019
        • Received: 1 July 2018
        Published in toct Volume 11, Issue 4

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