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 X ₁,..., 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, l} ^m which on an arbitrary (n, k)-bit-fixing source outputs m bits that are statistically-close to uniform. Recently, Kamp and Zuckerman [13] gave a construction of deterministic bit-fixing source extractor that extracts Ω(k ²/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.