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We propose a new model of a weakly random source that admits randomnessextraction. Our model of additive sources includes such natural sources asuniform distributions on arithmetic progressions (APs), generalized arithmeticprogressions (GAPs), and Bohr sets, each of which generalizes affine sources.We give an explicit extractor for additive sources with linear min-entropy overboth $\mathbb{Z}_p$ and $\mathbb{Z}_p^n$, for large prime $p$, although ourresults over $\mathbb{Z}_p^n$ require that the source further satisfy alist-decodability condition. As a corollary, we obtain explicit extractors forAPs, GAPs, and Bohr sources with linear min-entropy, although again our resultsover $\mathbb{Z}_p^n$ require the list-decodability condition. We furtherexplore special cases of additive sources. We improve previous constructions ofline sources (affine sources of dimension 1), requiring a field of size linearin $n$, rather than $\Omega(n^2)$ by Gabizon and Raz. This beats thenon-explicit bound of $\Theta(n \log n)$ obtained by the probabilistic method.We then generalize this result to APs and GAPs.

Deterministic Extractors for Additive Sources