Type of Document 
Master's Thesis 
Author 
McGee, John J.

URN 
etd04252006161727 
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 
20060425 
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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SchoofsAlgorithmThesisMcGee.pdf 
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ThesisMcGee06June2006.nb 
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