lec05.tex

\chapter{Algebraic Reconstruction I: The TZS Extractor}

\begin{enumerate}
  \item Define min-entropy.
  \item Define binary extractor, $q$-ary extractor. Extractor has two parameters: an $x$ with bounded min-entropy and a random $y$.
  \item The TZS $q$-ary extractor uses random input $y$ to choose $\vec{a} \in \F^2$ uniformly at random, encodes $x$ as a low-degree polynomial $\hat{x}$, and then outputs $\hat{x}(S(\vec{a})) \circ \hat{x}(S^2(\vec{a})) \circ \cdots \circ \hat{x}(S^m(\vec{a}))$.
  \item Proof idea: Reconstruction. If TZS is not a $q$-ary extractor, then by the hybrid argument, there is a predictor. We can use the predictor and a small amount of randomness to learn $\hat{x}$, and hence $\hat{x}$ has low entropy.
\end{enumerate}

\section{Extractors}

\section{The TZS Extractor}

\section{Sudan's Lemma}

\section*{The Upshot}

\begin{enumerate}
  \item Any reconstructive PRG is a sampler. The first step towards building a
    reconstructive PRG is to build a sampler.
\end{enumerate}