An oblivious bit-fixing source is a distribution over $\{0,1\}^n$, where $k$ bits are uniform and independent and the rest $n-k$ are fixed a priori to some constant value. Extracting (close to) true randomness from an oblivious bit-fixing source has been studied since the 1980s, with applications in cryptography and complexity theory.

We construct explicit extractors for oblivious bit-fixing source that support $k = \widetilde{O}(\log n)$, outputting almost all the entropy with low error. The previous state-of-the-art construction that outputs many bits is due to Rao [Rao, CCC '09], and require entropy $k \ge \log^{c}n$ for some large constant $c$. The two key components in our constructions are new low-error affine condensers for poly-logarithmic entropies (that we achieve using techniques from the nonmalleable extractors literature), and a dual use of linear condensers for OBF sources.