Generalised Veroneseans

Abstract

In [8], a characterization of the finite quadric Veronesean V 2 n n by means of properties of the set of its tangent spaces is proved. These tangent spaces form a 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.A typical problem in (finite) geometry is the study of highly symmetrical substructures. For example, arcs are configurations of points in PG(n, q) such that each n + 1 of them are in general position, while n 1 -dimensional dual arcs are sets of n 1 -spaces such that each two intersect in a point and any three of them are skew. These two structures appear naturally in cryptographical applications.In this article, we define objects, called (generalised) dual arcs; a class of objects that contain classical arcs and n 1 -dimensional dual arcs as special cases. These (generalised) dual arcs have applications in cryptography as well.We give a construction method for a wide class of parameters and prove an extension result for regular generalised dual arcs of order d = 1.In Sections 1 and 2, we give the necessary definitions, constructions, and examples of applications in cryptography. Section 3 refers to known classification results, and Section 4 states our main characterization theorem (Theorem 13). We now start with the required definitions to make this article self-contained.

Veroneseans, power subspaces and independence

“…23], [Sh,. Some of its geometric aspects have been studied outside Algebraic Geometry in [Lu,HT,CLS,BC,KSS1,KSS2].…”

“…, D ℓ ∈ ∆ and any subspace U that is an intersection of finitely many members of ∆. There are many stronger versions of this concept possible 2 , but this definition is geared to conform to the definitions appearing in [KSS1,KSS2].…”

“…Clearly the right side of ( 6.4) In [KSS2] there are objects with similar properties constructed without the use of polynomials. Note that D = A d .…”

“…Section 2.2 contains a surprisingly elementary proof. These types of results are in the geometric framework appearing in [Lu,HT,CLS,BC,KSS1,KSS2] rather than the more standard Algebraic Geometry occurrences of the Veronese map [Ha,p. 23], [Sh,.…”

“…, d, and the intersection of any d + 1 of them is 0. This notion was introduced in [KSS1,KSS2], but using projective dimension instead of vector space dimension. When d = 2 and n d = 1, D is a dual arc.…”