Algebraic methods in randomness and pseudorandomness

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 183-188).Algebra and randomness come together rather nicely in computation. A central example of this relationship in action is the Schwartz-Zippel lemma and its application to the fast randomized checking of polynomial identities. In this thesis, we further this relationship in two ways: (1) by compiling new algebraic techniques that are of potential computational interest, and (2) demonstrating the relevance of these techniques by making progress on several questions in randomness and pseudorandomness. The technical ingredients we introduce include: " Multiplicity-enhanced versions of the Schwartz-Zippel lenina and the "polynomial method", extending their applicability to "higher-degree" polynomials. " Conditions for polynomials to have an unusually small number of roots. " Conditions for polynomials to have an unusually structured set of roots, e.g., containing a large linear space. Our applications include: * Explicit constructions of randomness extractors with logarithmic seed and vanishing "entropy loss". " Limit laws for first-order logic augmented with the parity quantifier on random graphs (extending the classical 0-1 law). " Explicit dispersers for affine sources of imperfect randomness with sublinear entropy.by Swastik Kopparty.Ph.D

Extractors for Polynomial Sources over F2\mathbb{F}_2

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \frac{\sqrt{\log n}}{(d\log \log n)^{d/2}}$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction