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Approximate Degree, Secret Sharing, and Concentration Phenomena

Authors Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, Christopher Williamson

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  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • approximate degree
  • dual polynomial
  • pseudorandomness
  • polynomial approximation
  • secret sharing

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Abstract

The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are:
- We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution.
- We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena.
 As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1-norm at least K^{-3/2} * exp ({Omega (deg~_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND.

Cite As Get BibTex

Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, and Christopher Williamson. Approximate Degree, Secret Sharing, and Concentration Phenomena. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 71:1-71:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.71

Author Details

Andrej Bogdanov
  • Department of Computer Science and Engineering, Chinese University of Hong Kong
  • Institute for Theoretical Computer Science and Communications, Hong Kong
Nikhil S. Mande
  • Department of Computer Science, Georgetown University, USA
Justin Thaler
  • Department of Computer Science, Georgetown University, USA
Christopher Williamson
  • Department of Computer Science and Engineering, Chinese University of Hong Kong, Hong Kong

Funding

  • Bogdanov, Andrej: Supported by Hong Kong RGC GRF CUHK14207618.
  • Mande, Nikhil S.: Supported by NSF Grant CCF-1845125.
  • Thaler, Justin: Supported by NSF Grant CCF-1845125.
  • Williamson, Christopher: Supported by the Hong Kong PhD Fellowship Scheme.

Acknowledgements

We thank Mark Bun for telling us about the work of Sachdeva and Vishnoi [Sushant Sachdeva and Nisheeth K. Vishnoi, 2014], and Mert Sağlam, Pritish Kamath, Robin Kothari, and Prashant Nalini Vasudevan for helpful comments on a previous version of the manuscript. We are also grateful to Xuangui Huang and Emanuele Viola for sharing the manuscript [Xuangui Huang and Emanuele Viola, 2019].

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