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- W2947027153 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 error $epsilon$. The approximate degree of $f$ is at least $k$ iff there exists 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 new construction of a dual polynomial for the AND function, certifying that $deg_epsilon(f) geq Omega(sqrt{n log 1/epsilon})$. 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 read-once DNF is $Omega(sqrt{n})$. 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 error at most $K^{3/2} cdot exp(-Omega(k^2/K))$ for all $k < K < n/64$. This 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 analyses draw new connections between approximate degree and concentration phenomena. As a corollary, we show that for any $d deg_{1/3}(f)$. These upper and lower bounds were also previously only known in the case $f=$ AND." @default.
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- W2947027153 date "2019-06-02" @default.
- W2947027153 modified "2023-09-27" @default.
- W2947027153 title "Approximate degree, secret sharing, and concentration phenomena" @default.
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