A Note on Hypervector Spaces

The main aim of this paper is to generalize the concept of vector space by the hyperstructure. We generalize some definitions such as hypersubspaces, linear combination, Hamel basis, linearly dependence and linearly independence. A few important results like deletion theorem, extension theorem, dimension theorem have been established in this hypervector space.Comment: 17 page

Cubic Curves, Finite Geometry and Cryptography

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.Comment: 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

Hyperplanes of Hermitian dual polar spaces of rank 3 containing a quad

Let F and F' be two fields such that F' is a quadratic Galois extension of F. If vertical bar F vertical bar >= 3, then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space DH(5, F') to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of DH(5, q(2)), q not equal 2, arise from the Grassmann embedding, and to show that every hyperplane of DH(5, F') that contains a quad Q is either classical or the extension of a non-classical ovoid of Q. We will also give a classification of the hyperplanes of DH(5, F') that contain a quad and arise from the Grassmann embedding

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