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A Mathematica Package for Elliptic Curves

John McGee
Organization: Radford University

Wolfram Technology Conference 2015
Conference location

Champaign, Illinois USA

An elliptic curve is the solution set of a non-singular cubic equation in two unknowns. In general if F is a field and f is poly with degree(f)=3, such that f(x,y) and its partial derivatives do not vanish simultaneously then E={(x,y)|f(x,y)=0} is an elliptic curve. With so called ‘chord and tangent’ point addition, the set E becomes an abelian group. In this talk we present a Mathematica package for elliptic curve computations. The package includes special handling of elliptic curves over the reals, the complex numbers and over finite fields. For each of these fields we provide support for defining elliptic curves, generating random points on a curve, point addition P+Q, integer multiplication n P, along with computation of the j-Invariant. Over the reals the package provides support for graphical display of curves, point addition and support for finding rational points. Over the complex field the package supports use of the Weierstrass P-function and its derivative to map between an elliptic curve and its representation as torus. Over prime fields the package supports the determination of the order of a point and the order of the elliptic curve group along with cryptographic applications.

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