Approximate Degree, Secret Sharing, and Concentration Phenomena

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

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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


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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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)


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.

Subject Classification

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


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