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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.71
URN: urn:nbn:de:0030-drops-112869
URL: http://drops.dagstuhl.de/opus/volltexte/2019/11286/
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Bogdanov, Andrej ; Mande, Nikhil S. ; Thaler, Justin ; Williamson, Christopher

Approximate Degree, Secret Sharing, and Concentration Phenomena

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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.

BibTeX - Entry

@InProceedings{bogdanov_et_al:LIPIcs:2019:11286,
  author =	{Andrej Bogdanov and Nikhil S. Mande and Justin Thaler and Christopher Williamson},
  title =	{{Approximate Degree, Secret Sharing, and Concentration Phenomena}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{71:1--71:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11286},
  URN =		{urn:nbn:de:0030-drops-112869},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.71},
  annote =	{Keywords: approximate degree, dual polynomial, pseudorandomness, polynomial approximation, secret sharing}
}

Keywords: approximate degree, dual polynomial, pseudorandomness, polynomial approximation, secret sharing
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue Date: 2019
Date of publication: 19.09.2019


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