Deterministic extractors for bit-fixing sources by obtaining an independent seed

An (n, k)-bit-fixing source is a distribution X over {0,1}ⁿ such that there is a subset of k variables in X₁, ... , X_n which are uniformly distributed and independent of each other, and the remaining n - k variables are fixed. A deterministic bit-fixing source extractor is a function E : {0, 1}ⁿ → {0, 1}^m which on an arbitrary (n, k)-bit-fixing source outputs m bits that are statistically close to uniform. Recently, Kamp and Zuckerman [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 92-101] gave a construction of a deterministic bit-fixing source extractor that extracts ω(k² /n) bits and requires k > √n. In this paper we give constructions of deterministic bit-fixing source extractors that extract (1 - o(1))k bits whenever k > (log n)^c for some universal constant c > 0. Thus, our constructions extract almost all the randomness from bit-fixing sources and work even when k is small. For k ≫ √n the extracted bits have statistical distance 2^-nω(1) from uniform, and for k < √n the extracted bits have statistical distance k^-ω(1) from uniform. Our technique gives a general method to transform deterministic bit-fixing source extractors that extract few bits into extractors which extract almost all the bits.