Improved Randomness Extraction from Two Independent Sources

Given two independent weak random sources X,Y , with the same length l and min-entropies bX , bY whose sum is greater than l + Ω(polylog(l/ε)), we construct a deterministic two-source extractor (aka “blender”) that extracts max(bX , bY ) + (bX + bY − l − 4 log(1/ε)) bits which are ε-close to uniform. In contrast, best previously published construction [4] extracted at most 1 2 (bX + bY − l− 2 log(1/ε)) bits. Our main technical tool is a construction of a strong two-source extractor that extracts (bX + bY − l) − 2 log(1/ε) bits which are ε-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.