René Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Field

 dc.contributor.author McGee, John J. en dc.contributor.committeechair Brown, Ezra A. en dc.contributor.committeemember Parry, Charles J. en dc.contributor.committeemember Williams, Michael en dc.contributor.department Mathematics en dc.date.accessioned 2014-03-14T20:34:17Z en dc.date.adate 2006-06-08 en dc.date.available 2014-03-14T20:34:17Z en dc.date.issued 2006-04-25 en dc.date.rdate 2006-06-08 en dc.date.sdate 2006-04-25 en dc.description.abstract Elliptic curves have a rich mathematical history dating back to Diophantus (c. 250 C.E.), who used a form of these cubic equations to find right triangles of integer area with rational sides. In more recent times the deep mathematics of elliptic curves was used by Andrew Wiles et. al., to construct a proof of Fermat's last theorem, a problem which challenged mathematicians for more than 300 years. In addition, elliptic curves over finite fields find practical application in the areas of cryptography and coding theory. For such problems, knowing the order of the group of points satisfying the elliptic curve equation is important to the security of these applications. In 1985 René Schoof published a paper [5] describing a polynomial time algorithm for solving this problem. In this thesis we explain some of the key mathematical principles that provide the basis for Schoof's method. We also present an implementation of Schoof's algorithm as a collection of Mathematica functions. The operation of each algorithm is illustrated by way of numerical examples. en dc.description.degree Master of Science en dc.identifier.other etd-04252006-161727 en dc.identifier.sourceurl http://scholar.lib.vt.edu/theses/available/etd-04252006-161727/ en dc.identifier.uri http://hdl.handle.net/10919/31911 en dc.publisher Virginia Tech en dc.relation.haspart SchoofsAlgorithmThesisMcGee.pdf en dc.relation.haspart ThesisMcGee06June2006.nb en dc.rights In Copyright en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject Elliptic Curve en dc.subject Schoof en dc.subject Cryptography en dc.title René Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Field en dc.type Thesis en thesis.degree.discipline Mathematics en thesis.degree.grantor Virginia Polytechnic Institute and State University en thesis.degree.level masters en thesis.degree.name Master of Science en
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