Generalised dual arcs and Veronesean surfaces, with applications to cryptography

We start by defining generalised dual arcs, the motivation for defining them comes from cryptography, since they can serve as a tool to construct authentication codes and secret sharing schemes. We extend the characterisation of the tangent planes of the Veronesean surface V-2(4) in PG(5,q), q odd, described in [J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, New York, 1991], as a set of q(2) + q + 1 planes in PG(5, q), such that every two intersect in a point and every three are skew. We show that a set of q 2 + q planes generating PG(5, q), q odd, and satisfying the above properties can be extended to a set of q2 + q + I planes still satisfying all conditions. This result is a natural generalisation of the fact that a q-arc in PG(2, q), q odd, can always be extended to a (q + 1)-arc. This extension result is then used to study a regular generalised dual arc with parameters (9, 5, 2, 0) in PG(9, q), q odd, where we obtain an algebraic characterisation of such an object as being the image of a cubic Veronesean

Generalised Veroneseans

In \cite{ThasHVM}, a characterization of the finite quadric Veronesean $\mathcal{V}_{n}^{2^{n}}$ by means of properties of the set of its tangent spaces is proved. These tangent spaces form a {\em regular generalised dual arc}. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes

Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory