ECCC-Report TR19-079https://eccc.weizmann.ac.il/report/2019/079Comments and Revisions published for TR19-079en-usSun, 02 Jun 2019 10:05:18 +0300
Revision 2
| Average Bias and Polynomial Sources |
Arnab Bhattacharyya,
Philips George John,
Suprovat Ghoshal,
Raghu Meka
https://eccc.weizmann.ac.il/report/2019/079#revision2This paper has been withdrawn due to a mistake.Sun, 02 Jun 2019 10:05:18 +0300https://eccc.weizmann.ac.il/report/2019/079#revision2
Revision 1
| Average Bias and Polynomial Sources |
Arnab Bhattacharyya,
Philips George John,
Suprovat Ghoshal,
Raghu Meka
https://eccc.weizmann.ac.il/report/2019/079#revision1This paper has been withdrawn due to a mistake.Sun, 02 Jun 2019 10:04:27 +0300https://eccc.weizmann.ac.il/report/2019/079#revision1
Paper TR19-079
| Average Bias and Polynomial Sources |
Arnab Bhattacharyya,
Philips George John,
Suprovat Ghoshal,
Raghu Meka
https://eccc.weizmann.ac.il/report/2019/079We 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.Sat, 01 Jun 2019 19:39:21 +0300https://eccc.weizmann.ac.il/report/2019/079