{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/62425"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/62425","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Algebraic methods in randomness and pseudorandomness","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Kopparty, Swastik"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-22T22:21:25Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/62425","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Algebraic methods in randomness and pseudorandomness"]}]}],"canonical_facts":{"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."],"dc:creator":["Kopparty, Swastik"],"dc:date.accessioned":["2011-04-25T15:56:45Z"],"dc:date.available":["2011-04-25T15:56:45Z"],"dc:date.issued":["2010"],"dc:description":["Thesis (Ph. 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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."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/62425"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Electrical Engineering and Computer Science."],"dc:title":["Algebraic methods in randomness and pseudorandomness"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:25Z"}