Pseudorandomness, Symmetry, Smoothing: I

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



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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

Acknowledgements

CHL thanks Salil Vadhan and Terence Tao for helpful discussions.

Cite AsGet 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

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.

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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