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Browsing Pure Mathematics by Author "Barwick, S."
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Item Metadata only A class of Buekenhout unitals in the Hall plane(The Belgian Mathematical Society, 1996) Barwick, S.Let U be the classical unital in PG(2, q2) secant to ℓ∞. By deriving PG(2, q2) with respect to a derivation set disjoint from U we obtain a new unital U′ in the Hall plane of order q2. We show that this unital contains O'Nan configurations and is not isomorphic to the known unitals of the Hall plane, hence it forms a new class of unitals in the Hall plane.Item Metadata only A general approach to robust web metering(Kluwer Academic Publ, 2005) Barwick, S.; Jackson, W.; Martin, K.We consider the problem of metering access to web sites. Many services, such as web advertising, have a need for accurate counts of the number of visits to a web site. We consider the robust approach to web metering by looking at techniques that provide secure proofs of visit. We refine a number of previous models for metering schemes and provide a general construction for secure and efficient schemes that meter the interaction of a web site with a targeted audience. We generalise a technique for determining the minimum information that clients in the web site audience need to secure in order to supply proofs of visit. We also show how our metering schemes can be made robust against corrupt clients who attempt to prevent web servers constructing legitimate proofs of visit.Item Metadata only An optimal multisecret threshold scheme construction(Kluwer Academic Publ, 2005) Barwick, S.; Jackson, W.A multisecret threshold scheme is a system which protects a number of secret keys among a group of n participants. There is a secret sK associated with every subset K of k participants such that any t participants in K can reconstruct the secret sK, but a subset of w participants cannot get any information about a secret they are not associated with. This paper gives a construction for the parameters t = 2, k = 3 and for any n and w that is optimal in the sense that participants hold the minimal amount of information.Item Metadata only Flock generalized quadrangles and tetradic sets of elliptic quadrics of PG(3, q)(Academic Press Inc Elsevier Science, 2006) Barwick, S.; Brown, M.; Penttila, T.A flock of a quadratic cone of PG(3,q) is a partition of the non-vertex points into plane sections. It was shown by Thas in 1987 that to such flocks correspond generalized quadrangles of order (q2,q), previously constructed algebraically by Kantor (q odd) and Payne (q even). In 1999, Thas gave a geometrical construction of the generalized quadrangle from the flock via a particular set of elliptic quadrics in PG(3,q). In this paper we characterise these sets of elliptic quadrics by a simple property, construct the generalized quadrangle synthetically from the properties of the set and strengthen the main theorem of Thas 1999.Item Metadata only Generalising a characterisation of Hermitian curves(Birkhauser Verlag Ag, 2001) Barwick, S.; Quinn, C.This article proves a characterisation of the classical unital that is a generalisation of a characterisation proved in 1982 by Lefèvre-Percsy. It is shown that if U is a Buekenhout-Metz unital with respect to a line l∞ in PG(2, q²) such that a line of PG(2, q²) not through U Ո l∞ meets U in a Baer subline, then U is classical. An immediate corollary is that if U is a unital in PG(2, q²) such that U is Buekenhout-Metz with respect to two distinct lines, then U is classical.Item Metadata only Optimal linear perfect hash families with small parameters(John Wiley & Sons Inc, 2004) Barwick, S.; Jackson, W.; Quinn, C.A linear (qd, q, t)-perfect hash family of size s consists of a vector space V of order qd over a field F of order q and a sequence Φ1; . . . ; Φs of linear functions from V to F with the following property: for all t subsets X ⊆ V, there exists i ∈ {1; . . . ; s} such that Φi is injective when restricted to F. A linear (qd, q, t)--perfect hash family of minimal size d(t - 1) is said to be optimal. In this paper, we prove that optimal linear (qd, q, t)-perfect hash families exist only for q = 11 and for all prime powers q > 13 and we give constructions for these values of q.Item Metadata only Size of broadcast in threshold schemes with disenrollment(Springer-Verlag Berlin, 2002) Barwick, S.; Jackson, W.; Martin, K.; Wild, P.; Batten, L.; Seberry, J.Threshold schemes are well-studied cryptographic primitives for distributing information among a number of entities in such a way that the information can only be recovered if a threshold of entities co-operate. Establishment of a threshold scheme involves an initialisation overhead. Threshold schemes with disenrollment capability are threshold schemes that enable entities to be removed from the initial threshold scheme at less communication cost than that of establishing a new scheme. We prove a revised version of a conjecture of Blakley, Blakley, Chan and Massey by establishing a bound on the size of the broadcast information necessary in a threshold scheme with disenrollment capability that has minimal entity information storage requirements. We also investigate the characterisation of threshold schemes with disenrollment that meet this bound.Item Metadata only The Andre/Bruck and Bose representation in PG(2h,q): unitals and Baer subplanes(Belgian Mathematical Soc Triomphe, 2000) Barwick, S.; Casse, L.; Quinn, C.Many authors have used the André/Bruck and Bose representation of PG(2, q2) in PG(4, q) to study objects in the Desarguesian plane with great success. This paper looks at the André/Bruck and Bose representation of the Desarguesian plane PG(2, qh) in PG(2h, q) in order to determine whether this higher dimensional representation provides additional information about objects in the plane. In particular, we look at the representation of unitals and Baer subplanes in this setting.Item Metadata only The dual Yoshiara construction gives new extended generalized quadrangles(Academic Press Ltd Elsevier Science Ltd, 2004) Barwick, S.; Brown, M.A Yoshiara family is a set of q+3 planes in PG(5,q),q even, such that for any element of the set the intersection with the remaining q+2 elements forms a hyperoval. In 1998 Yoshiara showed that such a family gives rise to an extended generalized quadrangle of order (q+1,q−1). He also constructed such a family S(〇) from a hyperoval 〇 in PG(2,q). In 2000 Ng and Wild showed that the dual of a Yoshiara family is also a Yoshiara family. They showed that if 〇 has o-polynomial a monomial and 〇 is not regular, then the dual of S(〇) is a new Yoshiara family. This article extends this result and shows that in general the dual of S(〇) is a new Yoshiara family, thus giving new extended generalized quadrangles.Item Metadata only Unitals and Inversive Planes(Springer Science and Business Media LLC, 1997) Barwick, S.; O'Keefe, C.We show that if U is a Buekenhout-Metz unital (with respect to a point P) in any translation plane of order q2 with kernel containing GF(q), then U has an associated 2-(q2, q + 1, q) design which is the point-residual of an inversive plane, generalizing results of Wilbrink, Baker and Ebert. Further, our proof gives a natural, geometric isomorphism between the resulting inversive plane and the (egglike) inversive plane arising from the ovoid involved in the construction of the Buekenhout-Metz unital. We apply our results to investigate some parallel classes and partitions of the set of blocks of any Buekenhout-Metz unital. © Birkhäuser Verlag, Basel, 1997.Item Metadata only Unitals in the Hall Plane(Springer Science and Business Media LLC, 1997) Barwick, S.The unitals in the Hall plane are studied by deriving PG(2, q2) and observing the effect on the unitals of PG(2, q2). The number of Buekenhout and Buekenhout-Metz unitals in the Hall plane is determined. As a corollary we show that the classical unital is not embeddable in the Hall plane as a Buekenhout unital and that the Buekenhout unitals of H(q2) are not embeddable as Buekenhout unitals in the Desarguesian plane. Finally, we generalize this technique to other translation planes. © Birkhäuser Verlag, Basel, 1997.Item Metadata only Unitals which meet Baer subplanes in 1 modulo q points(Birkhauser Verlag Ag, 2000) Barwick, S.; O'Keefe, C.; Storme, L.We prove that a parabolic unital U in a translation plane π of order q2 with kernel containing GF(q) is a Buekenhout-Metz unital if and only if certain Baer subplanes containing the translation line of π meet U in 1 modulo q points. As a corollary we show that a unital U in PG(2, g2) is classical if and only if it meets each Baer subplane of PG(2,q2) in 1 modulo q points. © Birkhäuser Verlag, 2000.Item Open Access Updating the parameters of a threshold scheme by minimal broadcast(IEEE-Inst Electrical Electronics Engineers Inc, 2005) Barwick, S.; Jackson, W.; Martin, K.Threshold schemes allow secret data to be protected among a set of participants in such a way that only a prespecified threshold of participants can reconstruct the secret from private information (shares) distributed to them on a system setup using secure channels. We consider the general problem of designing unconditionally secure threshold schemes whose defining parameters (the threshold and the number of participants) can later be changed by using only public channel broadcast messages. In this paper, we are interested in the efficiency of such threshold schemes, and seek to minimize storage costs (size of shares) as well as optimize performance in low-bandwidth environments by minimizing the size of necessary broadcast messages. We prove a number of lower bounds on the smallest size of broadcast message necessary to make general changes to the parameters of a threshold scheme in which each participant already holds shares of minimal size. We establish the tightness of these bounds by demonstrating optimal schemes.