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

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

Background

Dispersers are weaker than extractors but play a central role in pseudorandomness. For sources sampled by AC^0[⊕] circuits or by low‑degree F2‑polynomial maps, even the existence of low‑complexity dispersers (let alone explicit ones) had been an open question.

This paper proves existence of low‑degree dispersers (non‑explicit) for several samplable source families using new bounds on Hilbert functions and degree‑d closures. Nonetheless, constructing explicit dispersers for AC^0[⊕] or low‑degree polynomial sources remains open.