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Generalised dual arcs and Veronesean surfaces, with applications to cryptography

Andreas Klein (UGent) , Jeroen Schillewaert (UGent) and Leo Storme (UGent)
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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.

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Please use this url to cite or link to this publication:

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.
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.
Chicago author-date
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.
Chicago author-date (all authors)
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.
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.
IEEE
[1]
A. Klein, J. Schillewaert, and L. Storme, “Generalised dual arcs and Veronesean surfaces, with applications to cryptography,” Journal of combinatorial theory, series A, vol. 116, no. 3, pp. 684–698, 2009.
@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},
}

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