Hard-to-Sample Distributions from Robust Extractors

TL;DR

This paper introduces a unified approach using robust extractors to construct explicit distributions that are hard for restricted computational models to generate, extending previous results and providing new hardness constructions.

Contribution

It develops a new notion of robust extractors to unify and extend hardness results for various sampling models, including low-depth circuits and low-degree polynomial sources.

Findings

Constructed explicit distributions with high distance from outputs of restricted models.

Extended Viola's framework to broader classes of sampling models.

Provided the first explicit distribution with high distance from low-degree polynomial sources.

Abstract

We provide a unified method for constructing explicit distributions which are difficult for restricted models of computation to generate. Our constructions are based on a new notion of robust extractors, which are extractors that remain sound even when a small number of points violate the min-entropy constraint. Using such objects, we show that for a broad range of sampling models (e.g., low-depth circuits, small-space sources, etc.), every output of the model has distance $1 - o(1)$ from our target distribution, qualitatively recovering essentially all previously known hardness results. Our work extends that of Viola (SICOMP '14), who developed an earlier unified framework based on traditional extractors to rule out sampling with very small error. As a further application of our technique, we leverage a recent extractor construction of Chattopadhyay, Goodman, and Gurumukhani (ITCS '24) to present the first explicit distribution with distance $1 - o(1)$ from the output of any low-degree $\mathbb{F}_2$-polynomial source. We note that a similar bound was obtained concurrently and independently by Khodabandeh and Shinkar (ECCC '26). We also describe a potential avenue toward proving a similar hardness result for $\mathsf{AC^0}[\oplus]$ circuits.