Deterministic Extractors for Bit-Fixing Sources by Obtaining an Independent Seed

Summary

An $(n,k)$-bit-fixing source is a distribution X over ${\{0,1\}}^n$ such that there is a subset of k variables in $X_1,\ldots,X_n$ 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 $E:{\{0,1\}}^n {\rightarrow} {\{0,1\}}^m$ which on an arbitrary $(n,k)$-bit-fixing source outputs m bits that are statistically-close to uniform. Prior to our work, Kamp and Zuckerman [44th FOCS, 2003] gave a construction of a deterministic bit-fixing source extractor that extracts $\Omega(k^2/n)$ bits and requires $k>\sqrt{n}$.

In this chapter 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 $k \gg \sqrt{n}$ the extracted bits have statistical distance $2^{-n^{\Omega(1)}}$ from uniform, and for $k \le \sqrt{n}$ the extracted bits have statistical distance $k^{-\Omega(1)}$ 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. This work is the first to use the ‘recycling paradigm’ as described in the introduction. The description of it here is different and perhaps more cumbersome, as the one given in the introduction was only realized in hindsight.

This chapter is based on [26].