### Online Linear Extractors for Independent Sources

Yevgeniy Dodis, Siyao Guo, Noah Stephens-Davidowitz, and Zhiye Xie

##### Abstract

In this work, we characterize online linear extractors. In other words, given a matrix $A \in \mathbb{F}_2^{n \times n}$, we study the convergence of the iterated process $\mathbf{S} \leftarrow A\mathbf{S} \oplus \mathbf{X}$, where $\mathbf{X} \sim D$ is repeatedly sampled independently from some fixed (but unknown) distribution $D$ with (min)-entropy at least $k$. Here, we think of $\mathbf{S} \in \{0,1\}^n$ as the state of an online extractor, and $\mathbf{X} \in \{0,1\}^n$ as its input. As our main result, we show that the state $\mathbf{S}$ converges to the uniform distribution for all input distributions $D$ with entropy $k > 0$ if and only if the matrix $A$ has no non-trivial invariant subspace (i.e., a non-zero subspace $V \subsetneq \mathbb{F}_2^n$ such that $AV \subseteq V$). In other words, a matrix $A$ yields an online linear extractor if and only if $A$ has no non-trivial invariant subspace. For example, the linear transformation corresponding to multiplication by a generator of the field $\mathbb{F}_{2^n}$ yields a good online linear extractor. Furthermore, for any such matrix convergence takes at most $\widetilde{O}(n^2(k+1)/k^2)$ steps. We also study the more general notion of condensing---that is, we ask when this process converges to a distribution with entropy at least $\ell$, when the input distribution has entropy greater than $k$. (Extractors corresponding to the special case when $\ell = n$.) We show that a matrix gives a good condenser if there are relatively few vectors $\mathbf{w} \in \mathbb{F}_2^n$ such that $\mathbf{w}, A^T\mathbf{w}, \ldots, (A^T)^{n-k-1} \mathbf{w}$ are linearly dependent. As an application, we show that the very simple cyclic rotation transformation $A(x_1,\ldots, x_n) = (x_n,x_1,\ldots, x_{n-1})$ condenses to $\ell = n-1$ bits for any $k > 1$ if $n$ is a prime satisfying a certain simple number-theoretic condition. Our proofs are Fourier-analytic and rely on a novel lemma, which gives a tight bound on the product of certain Fourier coefficients of any entropic distribution.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. MINOR revision.ITC 2021
DOI
10.4230/LIPIcs.ITC.2021.14
Keywords
extractorscondensorsentropy accumulationRNGs
Contact author(s)
noahsd @ gmail com
zx572 @ nyu edu
dodis @ cs nyu edu
siyao guo @ nyu edu
History
Short URL
https://ia.cr/2021/1002

CC BY

BibTeX

@misc{cryptoeprint:2021/1002,
author = {Yevgeniy Dodis and Siyao Guo and Noah Stephens-Davidowitz and Zhiye Xie},
title = {Online Linear Extractors for Independent Sources},
howpublished = {Cryptology ePrint Archive, Paper 2021/1002},
year = {2021},
doi = {10.4230/LIPIcs.ITC.2021.14},
note = {\url{https://eprint.iacr.org/2021/1002}},
url = {https://eprint.iacr.org/2021/1002}
}

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