Cubic Curves, Finite Geometry and Cryptography

Authors: Bruen, A.¹; Hirschfeld, J.²; Wehlau, D.³

Source: Acta Applicandae Mathematicae, Volume 115, Number 3, September 2011 , pp. 265-278(14)

Publisher: Springer

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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.

Keywords: Cubic curves; Group law; Non-singularity; Elliptic curve cryptography; Finite geometries

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10440-011-9620-z

Affiliations: 1: Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, T2N 1N4, Canada, Email: bruen@ucalgary.ca 2: Department of Mathematics, University of Sussex, Brighton, East Sussex, BN1 9RF, UK, Email: jwph@sussex.ac.uk 3: Department of Mathematics and Computer Science, Royal Military College, Kingston, Ontario, K7K 7B4, Canada, Email: wehlau@rmc.ca

Publication date: 2011-09-01