Explicit and low‑complexity extractors for AC^0[⊕] and low‑degree polynomial sources

Construct explicit deterministic extractors and determine the existence of non‑explicit low‑complexity extractors for distributions over {0,1}^n that are sampled by either AC^0[⊕] circuits or by low‑degree F2‑polynomial maps. The goal is to obtain nontrivial deterministic extractors (as opposed to trivial functions) and, in the non‑explicit regime, extractors computable by functions of low computational complexity, such as low‑degree polynomials or small circuits, for these source classes.

Background

The paper surveys the landscape of deterministic extraction from structured, samplable sources. While many classes admit either explicit or existential extractors, the situation for sources sampled by AC^0[⊕] circuits or by low‑degree F2‑polynomial maps is particularly challenging.

The authors note that prior to their work, even proving the existence of low‑complexity (e.g., low‑degree) extractors for these powerful source classes was unresolved, and explicit constructions were also unknown. This paper establishes existence of low‑degree extractors (non‑explicit) for several samplable families, advancing the state of knowledge on the low‑complexity front. However, constructing explicit deterministic extractors for AC^0[⊕] or low‑degree polynomial sources remains open.