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, l}ⁿ → {0, l}^m which on an arbitrary (n, k)-bit-fixing source outputs m bits that are statistically-close to uniform. Recently, Kamp and Zuckerman (2003) gave a construction of 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.