Title page for ETD etd-04252006-161727


Type of Document Master's Thesis
Author McGee, John J.
URN etd-04252006-161727
Title René Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Field
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Brown, Ezra A. Committee Chair
Parry, Charles J. Committee Member
Williams, Michael Committee Member
Keywords
  • Elliptic Curve
  • Schoof
  • Cryptography
Date of Defense 2006-04-25
Availability unrestricted
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.
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