# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.11612