Low-Degree Polynomials Are Good Extractors

We prove that random low-degree polynomials (over $\mathbb{F}_2$) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, (2) affine sources, and (3) local sources. We significantly generalize these results, and prove the following. 1. Low-degree polynomials extract from small families. We show that a random low-degree polynomial is a good low-error extractor for any small family of sources. In particular, we improve the positive result of Alrabiah, Chattopadhyay, Goodman, Li, and Ribeiro (ICALP 2022) for local sources, and give new results for polynomial sources and variety sources via a single unified approach. 2. Low-degree polynomials extract from sumset sources. We show that a random low-degree polynomial is a good extractor for sumset sources, which are the most general large family of sources (capturing independent sources, interleaved sources, small-space sources, and more). This extractor achieves polynomially small error, and its min-entropy requirement is tight up to a square. Our results on sumset extractors imply new complexity separations for linear ROBPs, and the tools that go into its proof have further applications, as well. The two main tools we use are a new structural result on sumset-punctured Reed-Muller codes, paired with a novel type of reduction between randomness extractors. Using the first new tool, we strengthen and generalize the extractor impossibility results of Chattopadhyay, Goodman, and Gurumukhani (ITCS 2024). Using the second, we show the existence of sumset extractors for min-entropy $k=O(\log(n/\varepsilon))$, resolving an open problem of Chattopadhyay and Liao (STOC 2022).