Cubic Curves, Finite Geometry and Cryptography

Authors:
A. A. Bruen;J. W. Hirschfeld;D. L. Wehlau
Affiliations:
Department of Electrical and Computer Engineering, University of Calgary, Calgary, Canada T2N 1N4;Department of Mathematics, University of Sussex, Brighton, UK BN1 9RF;Department of Mathematics and Computer Science, Royal Military College, Kingston, Canada K7K 7B4
Venue:
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Year:
2011

Citing 4
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Abstract

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.