Generalised dual arcs and Veronesean surfaces, with applications to cryptography

Abstract

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 q 2 + q + 1 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.