Affine Extractors for Almost Logarithmic Entropy

We give an explicit construction of an affine extractor (over$\mathbb{F}_{2}$) that works for affine sources on$n$bits with min-entropy$k\geq\log n\cdot(\log\log n)^{1+o(1)}$. This improves prior work of Li (FOCS'16) that requires min-entropy at least$\text{poly} (\log n)$. Our construction is based on the framework of using correlation breakers and resilient functions, a paradigm that was also used by Li. On a high level, the key sources of our improvement are based on the following new ingredients: (i) A new construction of an affine somewhere random extractor, that we use in a crucial step instead of a linear seeded extractor (for which optimal constructions are not known) that was used by Li. (ii) A near optimal construction of a correlation breaker for linearly correlated sources. The construction of our correlation breaker takes inspiration from an exciting line of recent work that constructs two-source extractors for near logarithmic min-entropy.