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

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.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2006

Author dc:creator

• McGee, John J.

Chair dc:contributor.committeechair

• Brown, Ezra A.

Committee members dc:contributor.committeemember

• Parry, Charles J.
• Williams, Michael

• Elliptic Curve
• Schoof
• Cryptography

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-04252006-161727

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/31911