Linear Sections of the Finite Veronese Varieties and Authentication Systems Defined Using Geometry

Abstract

In this paper we deal with authentication systems in which one key is used to authenticate many source states. We answer a related question on the cardinalities of the intersections of quadrics in PG (d,q). We first generalize a class of geometric authentication systems, which has been introduced by Beutelspacher, Tallini and Zanella4. The source states are the lines through a special point N of PG (d,q) (the d-dimensional projective space over GF (q)). The keys are some hypersurfaces which have N as a nucleus ( N is a nucleus of Σ if every line through N meets Σ in exactly one point). The message belonging to a source state ℓ and a key Σ is the unique point of intersection of the line ℓ with the hypersurface Σ. We give the values of s for which the constructed authentication systems have a security which is comparable to the best allowed by a theoretical bound. In case the hypersurfaces are quadrics, we give further results on the security. To this end, we determine the greatest cardinality for the intersections of the finite Veronese varieties with the projective subspaces of any given dimension. Finally, we discuss a possible implementation.