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

Determine whether there exist nontrivial explicit deterministic dispersers and, more generally, non-explicit low-complexity dispersers 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

Dispersers are weaker than extractors: they only require that the output not be constant on the source’s support. For powerful samplable sources such as AC^0[⊕] circuits and low-degree F2-polynomial maps, even this relaxed objective had not been achieved explicitly, and the authors note that the corresponding existence results for low-complexity dispersers were also open.

The paper later proves existence of low-degree dispersers for several source families using new Hilbert-function bounds, but the introduction explicitly records that, at the time of framing, the analogous problems for AC^0[⊕] and low-degree polynomial sources were open.