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Curves and Their Applications to Factoring Polynomials

dc.contributor.advisorWashington, Lawrence Cen_US
dc.contributor.authorOzdemir, Enveren_US
dc.date.accessioned2009-10-21T05:30:13Z
dc.date.available2009-10-21T05:30:13Z
dc.date.issued2009en_US
dc.identifier.urihttp://hdl.handle.net/1903/9689
dc.description.abstractWe present new methods for computing square roots and factorization of polynomials over finite fields. We also describe a method for computing in the Jacobian of a singular hyperelliptic curve. There is a compact representation of an element in the Jacobian of a smooth hyperelliptic curve over any field. This compact representation leads an efficient method for computing in Jacobians which is called Cantor's Algorithm. In one part of the dissertation, we show that an extension of this compact representation and Cantor's Algorithm is possible for singular hyperelliptic curves. This extension lead to the use of singular hyperelliptic curves for factorization of polynomials and computing square roots in finite fields. Our study shows that computing the square root of a number mod p is equivalent to finding any of the particular group elements in the Jacobian of a certain singular hyperelliptic curve. This is also true in the case of polynomial factorizations. Therefore the efficiency of our algorithms depends on only the efficiency of the algorithms for computing in the Jacobian of a singular hyperelliptic curve. The algorithms for computing in Jacobians of hyperelliptic curves are very fast especially for small genus and this makes our algorithms especially computing square roots algorithms competitive with the other well-known algorithms. In this work we also investigate superelliptic curves for factorization of polynomials.en_US
dc.format.extent307284 bytes
dc.format.mimetypeapplication/pdf
dc.language.isoen_US
dc.titleCurves and Their Applications to Factoring Polynomialsen_US
dc.typeDissertationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.contributor.departmentMathematicsen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledFactoring Polynomialsen_US
dc.subject.pquncontrolledHyperelliptic Curvesen_US
dc.subject.pquncontrolledJacobianen_US
dc.subject.pquncontrolledPicard Groupen_US


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