We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, this model was introduced by De and Watson (TOCT 2012) and Viola (SICOMP 2014), and is closely related to sources generated by $\mathsf{AC}^0$ circuits and bounded-width branching programs. In particular, extractors for local sources also work for sources generated by these classical computational models. Despite being introduced a decade ago, little progress has been made on improving the entropy requirement for extracting from local sources. The current best explicit extractors require entropy $n^{1/2}$, and follow via a reduction to affine extractors. To start, we prove a barrier showing that one cannot hope to improve this entropy requirement via a black-box reduction of this form. In particular, new techniques are needed. In our main result, we seek to answer whether low-degree polynomials (over $\mathbb{F}_2$) hold potential for breaking this barrier. We answer this question in the positive, and fully characterize the power of low-degree polynomials as extractors for local sources. More precisely, we show that a random degree $r$ polynomial is a low-error extractor for $n$-bit local sources with min-entropy $\Omega(r(n\log n)^{1/r})$, and we show that this is tight. Our result leverages several new ingredients, which may be of independent interest. Our existential result relies on a new reduction from local sources to a more structured family, known as local non-oblivious bit-fixing sources. To show its tightness, we prove a "local version" of a structural result by Cohen and Tal (RANDOM 2015), which relies on a new "low-weight" Chevalley-Warning theorem.
翻译:我们继续从由简单进程生成的微弱源中随机提取比特。 我们继续一项从微弱源中提取随机比特的工作。 特别是, 本地源的提取器也为这些古典计算模型生成的来源工作。 尽管在十年前引入了源中, 但对于从本地源中提取的微小( 隐藏的) 任意随机比特输入比特。 另外, De and Watson ( TOCT 2012) 和 Viola ( SICOMP 2014) 引入了这个模型, 并且与美元和美元( SICOMP 2014 ) 生成的微弱源密切相关。 首先, 我们证明一个屏障, 无法通过黑盒减少这种形式的微小分量需求。 特别是, 本地源的提取器也为这些古典计算模型生成的源。 尽管在十年前引入了这些源中, 在改进从本地源中提取的最小值要求方面进展甚微。 当前的最明显提取器需要的是, 我们的低度阻断层的硬度 硬度的根值源值 。