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Elliptic Curve Algorithms in Mathematica

John McGee
Organization: Radford University

2007 Wolfram Technology Conference
Conference location

Champaign, IL


Elliptic curves are nonsingular polynomials of degree three in two variables, as members of F[x,y]. Points on the graph of an elliptic curve can be combined using a special addition operator to turn the graph into an Abelian group. When F is a finite field, these curves are applied to problems and algorithms in cryptography and number theory. Determining the order of the group of points on an elliptic curve over a finite field is an important subproblem in many of these applications. We present a Mathematica implementation of René Schoof's algorithm for determining this group order along with examples of its use. Our implementation leverages Mathematica's number theoretic and polynomial algebra capabilities. We will also describe how Mathematica was helpful in verifying the equations that provide the basis for this algorithm.

*Wolfram Technology

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EllipticCurveAlgorithms.nb (1.6 MB) - Mathematica Notebook [for Mathematica 6.0]