Showing papers by "J. W. P. Hirschfeld published in 2011"
TL;DR: Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed and a possible strengthening of the security of elliptic curve cryptography is proposed using a ‘shared secret’ related to the group law.
Abstract: Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.
11 citations
Book•
17 Sep 2011
TL;DR: In this paper, the authors present a graduate-level book aimed at experienced researchers as well as beginning graduate students in finite geometry, incidence geometry, design theory, coding theory and combinatorics.
Abstract: This is a graduate-level book aimed at experienced researchers as well as beginning graduate students in finite geometry, incidence geometry, design theory, coding theory and combinatorics. The 21 articles arise from the Fourth Isle of Thorns Conference on Finite Geometries, held in July 2000. The book contains new high-level results at the forefront of the field as well as authoritative surveys. The articles are written so that beginning graduate students can appreciate them. Among the surveys by the invited speakers are P.J. Cameron `Fixed Points and cycles', C.E. Praeger `Implications of line-transitivity for designs', B. Schmidt, `Exponent bounds', which explains recent results on difference sets, and several papers with H. Van Maideghem as co-author on generalized polygons. The editors also contribute: A. Blokhuis, D. Jungnickel, and B. Schmidt `On a class of symmetric divisible designs which are almost projective planes', and D. Luyckx and J.A. Thas discuss the geometry of the quadric in six dimensions. There is also a lengthy and complete survey by J.W.P. Hirschfeld and L. Storme `The packing problem in statistics, coding theory and finite projective spaces: update 2001'. Audience: Researchers as well as beginning graduate students in the areas of finite geometry, incidence geometry, design theory, coding theory and combinatorics will appreciate the surveys and original papers by leaders in these fields.
4 citations
TL;DR: In this article, the security of non-singular cubic curves over finite fields has been studied and a possible strengthening of elliptic curve cryptography is proposed using a ''shared secret'' related to the group law.
Abstract: Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.
1 citations