Approximate Degree, Secret Sharing, and Concentration Phenomena
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.
approximate degree
dual polynomial
pseudorandomness
polynomial approximation
secret sharing
Theory of computation~Pseudorandomness and derandomization
71:1-71:21
RANDOM
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/082/.
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].
Andrej
Bogdanov
Andrej Bogdanov
Department of Computer Science and Engineering, Chinese University of Hong Kong
Institute for Theoretical Computer Science and Communications, Hong Kong
Supported by Hong Kong RGC GRF CUHK14207618.
Nikhil S.
Mande
Nikhil S. Mande
Department of Computer Science, Georgetown University, USA
Supported by NSF Grant CCF-1845125.
Justin
Thaler
Justin Thaler
Department of Computer Science, Georgetown University, USA
Supported by NSF Grant CCF-1845125.
Christopher
Williamson
Christopher Williamson
Department of Computer Science and Engineering, Chinese University of Hong Kong, Hong Kong
Supported by the Hong Kong PhD Fellowship Scheme.
10.4230/LIPIcs.APPROX-RANDOM.2019.71
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Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, and Christopher Williamson
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