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Pseudorandomness, Symmetry, Smoothing: I

Authors Harm Derksen, Peter Ivanov, Chin Ho Lee , Emanuele Viola

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Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • pseudorandomness
  • k-wise uniform distributions
  • small-bias distributions
  • noise
  • symmetric tests
  • thresholds
  • Krawtchouk polynomials

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Abstract

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that  
- achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. 
- have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 
- rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley.  Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.

Cite As Get BibTex

Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola. Pseudorandomness, Symmetry, Smoothing: I. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 18:1-18:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.18

Author Details

Harm Derksen
  • Northeastern University, Boston, MA, USA
Peter Ivanov
  • Northeastern University, Boston, MA, USA
Chin Ho Lee
  • North Carolina State University, Raleigh, NC, USA
Emanuele Viola
  • Northeastern University, Boston, MA, USA

Funding

  • Derksen, Harm: Partially supported by NSF grant DMS 2147769.
  • Ivanov, Peter: Supported by NSF grant CCF-2114116.
  • Viola, Emanuele: Supported by NSF grant CCF-2114116.

Acknowledgements

CHL thanks Salil Vadhan and Terence Tao for helpful discussions.

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