Average Bias and Polynomial Sources

TL;DR

This paper introduces the concept of average bias as a stronger measure than min-entropy for randomness sources, especially polynomial sources, and demonstrates its implications for extractors and dispersers.

Contribution

It defines average bias, explores its relation to min-entropy for polynomial sources, and designs dispersers for quadratic sources based on this new notion.

Findings

Inner product function extracts randomness from sources with low average bias.

For quadratic sources, min-entropy implies an average bias of at most 2^{- ext{Omega}(\sqrt{k})}.

Dispersers are constructed for quadratic sources with min-entropy guarantees.

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: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim Z}(-1)^{\langle c, z\rangle}|$. 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.