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Revision #1 to TR21-147 | 25th October 2021 08:27

#### Extractors for Sum of Two Sources

Revision #1
Accepted on: 25th October 2021 08:27
Keywords:

Abstract:

We consider the problem of extracting randomness from \textit{sumset sources}, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An $(n,k,C)$-sumset source $\mathbf{X}$ is a distribution on $\{0,1\}^n$ of the form $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_C$, where $\mathbf{X}_i$'s are independent sources on $n$ bits with min-entropy at least $k$. Prior extractors either required the number of sources $C$ to be a large constant or the min-entropy $k$ to be at least $0.51 n$.

As our main result, we construct an explicit extractor for sumset sources in the setting of $C=2$ for min-entropy $\mathrm{poly}(\log n)$ and polynomially small error. We can further improve the min-entropy requirement to $(\log n) \cdot (\log \log n)^{1 + o(1)}$ at the expense of worse error parameter of our extractor. We find applications of our sumset extractor for extracting randomness from other well-studied models of weak sources such as affine sources, small-space sources, and interleaved sources.

Interestingly, it is unknown if a random function is an extractor for sumset sources. We use techniques from additive combinatorics to show that it is a disperser, and further prove that an affine extractor works for an interesting subclass of sumset sources which informally corresponds to the low doubling" case
(i.e., the support of $\mathbf{X_1} + \mathbf{X_2}$ is not much larger than $2^k$).

Changes to previous version:

fixed some typos

### Paper:

TR21-147 | 22nd October 2021 04:22

#### Extractors for Sum of Two Sources

TR21-147
Publication: 24th October 2021 14:46
We consider the problem of extracting randomness from \textit{sumset sources}, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An $(n,k,C)$-sumset source $\mathbf{X}$ is a distribution on $\{0,1\}^n$ of the form $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_C$, where $\mathbf{X}_i$'s are independent sources on $n$ bits with min-entropy at least $k$. Prior extractors either required the number of sources $C$ to be a large constant or the min-entropy $k$ to be at least $0.51 n$.
As our main result, we construct an explicit extractor for sumset sources in the setting of $C=2$ for min-entropy $\mathrm{poly}(\log n)$ and polynomially small error. We can further improve the min-entropy requirement to $(\log n) \cdot (\log \log n)^{1 + o(1)}$ at the expense of worse error parameter of our extractor. We find applications of our sumset extractor for extracting randomness from other well-studied models of weak sources such as affine sources, small-space sources, and interleaved sources.