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

Determine whether there exist nontrivial explicit deterministic extractors and, more generally, non-explicit low-complexity extractors for distributions on {0,1}^n that are sampled by either AC^0[⊕] circuits or by low-degree polynomials over F2 (i.e., sources of the form f(U_m) where f is an AC^0[⊕] circuit or a low-degree F2-polynomial).

Background

Randomness extraction from samplable sources is a central topic in pseudorandomness. While many results are known for weaker classes (e.g., local or decision-tree sources), handling more powerful generators such as AC^0[⊕] circuits or low-degree F2-polynomial maps has proven difficult. The paper highlights that even obtaining nontrivial explicit extractors for these families has remained elusive.

This work develops tools via Hilbert function lower bounds and degree-d closures, proving existence of low-degree extractors for several samplable families. However, the authors explicitly note that, for AC^0[⊕] and low-degree polynomial sources, constructing nontrivial explicit extractors and even establishing the existence of non-explicit low-complexity extractors was open at the time they set up the problem statement.