← 研踪 PaperTrailarXiv:2611.07266 · v1
Quadratic Extractors
首次提交 2026-11-05 08:36
版本更新 2026-11-05 08:36
雷达观测 2026-11-06 10:04
摘要
A quadratic source is a random variable $X\sim\mathbb{F}_2^n$ generated by applying $n$ quadratics to a hidden uniform string. Previously, the best-known explicit extractor for this model required min-entropy roughly $k=n-\sqrt{\log n}$. In this paper, we give an explicit construction for min-entropy $k=\mathrm{polylog}(n)$. Even for the special case of $2$-local sources, where each bit of $X$ depends on two hidden uniform bits, achieving an entropy requirement below $k=\sqrt{n}$ was an important open problem. This is because such sources provably cannot be reduced (in a black-box manner) to our most powerful general-purpose seedless extractor, known as a sumset extractor. Our new extractor bypasses this barrier, providing the first explicit construction to go beyond sumset extraction.
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v1 · 2026-11-05 08:36 · 最新观测版本
Quadratic Extractors
A quadratic source is a random variable $X\sim\mathbb{F}_2^n$ generated by applying $n$ quadratics to a hidden uniform string. Previously, the best-known explicit extractor for this model required min-entropy roughly $k=n-\sqrt{\log n}$. In this paper, we give an explicit construction for min-entropy $k=\mathrm{polylog}(n)$. Even for the special case of $2$-local sources, where each bit of $X$ depends on two hidden uniform bits, achieving an entropy requirement below $k=\sqrt{n}$ was an important open problem. This is because such sources provably cannot be reduced (in a black-box manner) to our most powerful general-purpose seedless extractor, known as a sumset extractor. Our new extractor bypasses this barrier, providing the first explicit construction to go beyond sumset extraction.
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