Improved Randomness Extraction from Two Independent Sources

Given two independent weak random sources X,Y, with the same length l and min-entropies b X , b Y whose sum is greater than \(\ell+ \Omega(\mbox{\sf polylog}(\ell/\varepsilon))\), we construct a deterministic two-source extractor (aka “blender”) that extracts max (b X ,b Y ) + (b X + b Y − − l − − 4log(1/e)) bits which are e-close to uniform. In contrast, best previously published construction [4] extracted at most \(\frac{1}{2}(b_X + b_Y -- \ell -- 2\log(1/\varepsilon))\) bits. Our main technical tool is a construction of a strong two-source extractor that extracts (b X + b Y – l) – 2log(1/e) bits which are e-close to being uniform and independent of one of the sources (aka “strong blender”), so that they can later be reused as a seed to a seeded extractor. Our strong two-source extractor construction improves the best previously published construction of such strong blenders [7] by a factor of 2, applies to more sources X and Y, and is considerably simpler than the latter. Our methodology also unifies several of the previous two-source extractor constructions from the literature.

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