Deterministic Extractors for Additive Sources

We propose a new model of a weakly random source that admits randomness extraction. Our model of additive sources includes such natural sources as uniform distributions on arithmetic progressions (APs), generalized arithmetic progressions (GAPs), and Bohr sets, each of which generalizes affine sources. We give an explicit extractor for additive sources with linear minentropy over both Zp and Z n p , for large prime p, although our results over Z p require that the source further satisfy a list-decodability condition. As a corollary, we obtain explicit extractors for APs, GAPs, and Bohr sources with linear min-entropy, although again our results over Z p require the list-decodability condition. We further explore special cases of additive sources. We improve previous constructions of line sources (affine sources of dimension 1), requiring a field of size linear in n, rather than Ω(n) by Gabizon and Raz. This beats the non-explicit bound of Θ(n logn) obtained by the probabilistic method. We then generalize this result to APs and GAPs. [email protected], Department of Computer Science, The University of Texas at Austin. Research supported in part by NSF Grants CCF-0916160 and CCF-1218723. [email protected], Computer Science Department, Technion, Haifa, Israel. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575. [email protected], Department of Mathematics, The University of Texas at Austin. [email protected], Department of Computer Science, The University of Texas at Austin. Research supported in part by NSF Grants CCF-0916160 and CCF-1218723.