Average Bias and Polynomial Sources

Abstract

We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1{\}}^n$, its average bias is: $b_{\text{av}}(Z)=2^{-n}{\sum }_{c\in \{0,1{\}}^n}|{\mathbb{E}}_{z\sim Z}(-1{)}^{⟨c,z⟩}|$. A source with average bias at most $2^{-k}$ has min-entropy at least $k$, and so low average bias is a stronger condition than high min-entropy. We observe that the inner product function is an extractor for any source with average bias less than $2^{-n/2}$. The notion of average bias especially makes sense for polynomial sources, i.e., distributions sampled by low-degree $n$-variate polynomials over ${\mathbb{F}}_2$. For the well-studied case of affine sources, it is easy to see that min-entropy $k$ is exactly equivalent to average bias of $2^{-k}$. We show that for quadratic sources, min-entropy $k$ implies that the average bias is at most $2^{-\Omega (\sqrt{k})}$. We use this relation to design dispersers for separable quadratic sources with a min-entropy guarantee.

We found out one of the main results has a much easier and direct proof