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Generalised dual arcs and Veronesean surfaces, with applications to cryptography

Andreas Klein UGent, Jeroen Schillewaert UGent and Leo Storme UGent (2009) Journal of combinatorial theory, series A. 116(3). p.684-698
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 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.
Please use this url to cite or link to this publication:
author
organization
year
type
journalArticle (original)
publication status
published
subject
journal title
Journal of combinatorial theory, series A
J. Combin. Theory, Ser. A
volume
116
issue
3
pages
684 - 698
Web of Science type
Article
Web of Science id
000264406900011
JCR category
MATHEMATICS
JCR impact factor
0.783 (2009)
JCR rank
81/251 (2009)
JCR quartile
2 (2009)
ISSN
0097-3165
DOI
10.1016/j.jcta.2008.11.001
language
English
UGent publication?
yes
classification
A1
id
592357
handle
http://hdl.handle.net/1854/LU-592357
date created
2009-04-09 11:30:22
date last changed
2009-04-10 14:02:45
@article{592357,
  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 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.},
  author       = {Klein, Andreas and Schillewaert, Jeroen and Storme, Leo},
  issn         = {0097-3165},
  journal      = {Journal of combinatorial theory, series A},
  language     = {eng},
  number       = {3},
  pages        = {684--698},
  title        = {Generalised dual arcs and Veronesean surfaces, with applications to cryptography},
  url          = {http://dx.doi.org/10.1016/j.jcta.2008.11.001},
  volume       = {116},
  year         = {2009},
}

Chicago
Klein, Andreas, Jeroen Schillewaert, and Leo Storme. 2009. “Generalised Dual Arcs and Veronesean Surfaces, with Applications to Cryptography.” Journal of Combinatorial Theory, Series A 116 (3): 684–698.
APA
Klein, A., Schillewaert, J., & Storme, L. (2009). Generalised dual arcs and Veronesean surfaces, with applications to cryptography. Journal of combinatorial theory, series A, 116(3), 684–698.
Vancouver
1.
Klein A, Schillewaert J, Storme L. Generalised dual arcs and Veronesean surfaces, with applications to cryptography. Journal of combinatorial theory, series A. 2009;116(3):684–98.
MLA
Klein, Andreas, Jeroen Schillewaert, and Leo Storme. “Generalised Dual Arcs and Veronesean Surfaces, with Applications to Cryptography.” Journal of combinatorial theory, series A 116.3 (2009): 684–698. Print.