Better lossless condensers through derandomized curve samplers

Lossless condensers are unbalanced expander graphs, with expansion close to optimal. Equivalently, they may be viewed as functions that use a short random seed to map a source on n bits to a source on many fewer bits while preserving all of the min-entropy. It is known how to build lossless condensers when the graphs are slightly unbalanced in the work of M. Capalbo et al. (2002). The highly unbalanced case is also important but the only known construction does not condense the source well. We give explicit constructions of lossless condensers with condensing close to optimal, and using near-optimal seed length. Our main technical contribution is a randomness-efficient method for sampling F^D (where F is a field) with low-degree curves. This problem was addressed before in the works of E. Ben-Sasson et al. (2003) and D. Moshkovitz and R. Raz (2006) but the solutions apply only to degree one curves, i.e., lines. Our technique is new and elegant. We use sub-sampling and obtain our curve samplers by composing a sequence of low-degree manifolds, starting with high-dimension, low-degree manifolds and proceeding through lower and lower dimension manifolds with (moderately) growing degrees, until we finish with dimension-one, low-degree manifolds, i.e., curves. The technique may be of independent interest