Cubic Curves, Finite Geometry and Cryptography
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
Cite
@article{arxiv.1107.4387,
title = {Cubic Curves, Finite Geometry and Cryptography},
author = {A. A. Bruen and J. W. P. Hirschfeld and D. L. Wehlau},
journal= {arXiv preprint arXiv:1107.4387},
year = {2011}
}
Comments
This is a version of our article to appear in Acta Applicandae Mathematicae. In this version, we have corrected a sentence in the third paragraph. The final publication is available at springerlink.com at http://www.springerlink.com/content/xh85647871215644/