eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-09-17
71:1
71:21
10.4230/LIPIcs.APPROX-RANDOM.2019.71
article
Approximate Degree, Secret Sharing, and Concentration Phenomena
Bogdanov, Andrej
1
2
Mande, Nikhil S.
3
Thaler, Justin
3
Williamson, Christopher
4
Department of Computer Science and Engineering, Chinese University of Hong Kong
Institute for Theoretical Computer Science and Communications, Hong Kong
Department of Computer Science, Georgetown University, USA
Department of Computer Science and Engineering, Chinese University of Hong Kong, Hong Kong
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol145-approx-random2019/LIPIcs.APPROX-RANDOM.2019.71/LIPIcs.APPROX-RANDOM.2019.71.pdf
approximate degree
dual polynomial
pseudorandomness
polynomial approximation
secret sharing