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

dc.contributor.authorMcGee, John J.en
dc.contributor.committeechairBrown, Ezra A.en
dc.contributor.committeememberParry, Charles J.en
dc.contributor.committeememberWilliams, Michaelen
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:34:17Zen
dc.date.adate2006-06-08en
dc.date.available2014-03-14T20:34:17Zen
dc.date.issued2006-04-25en
dc.date.rdate2006-06-08en
dc.date.sdate2006-04-25en
dc.description.abstractElliptic 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.degreeMaster of Scienceen
dc.identifier.otheretd-04252006-161727en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-04252006-161727/en
dc.identifier.urihttp://hdl.handle.net/10919/31911en
dc.publisherVirginia Techen
dc.relation.haspartSchoofsAlgorithmThesisMcGee.pdfen
dc.relation.haspartThesisMcGee06June2006.nben
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectElliptic Curveen
dc.subjectSchoofen
dc.subjectCryptographyen
dc.titleRené Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Fielden
dc.typeThesisen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.levelmastersen
thesis.degree.nameMaster of Scienceen
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