Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.

Contemporary take on deconstructed Medea : a review

THEATRE: The New Dead: Medea Material. By Heiner Muller. Stella Electrika in association with La Boite Theatre Company, Brisbane, November 19. THERE has been a lot of intensity in independent theatre in Brisbane during the past year, as companies, production houses and producers have begun building new programs and platforms to support an expansion of pathways within the local theatre ecology. Audiences have been exposed to works signalling the diversity of what Brisbane theatre makers want to see on stage, from productions of new local and international pieces to new devised works, and the results of residencies and development programs. La Boite Theatre Company closes its inaugural indie season with a work that places it at the contemporary, experimental end of the spectrum. The New Dead: Medea Material is emerging director Kat Henry's interpretation of Heiner Muller's 1981 text Despoiled Shore Medea Material Landscape with Argonauts. Start of sidebar. Skip to end of sidebar. End of sidebar. Return to start of sidebar. Muller is known for his radical adaptations of historical dramas, from the Greeks to Shakespeare, and for deconstructed texts in which the characters - in this case, Medea - violently reject the familial, cultural and political roles society has laid out for them. Muller's combination of deconstructed characters, disconnected poetic language and constant references to aspects of popular culture and the Cold War politics he sought to abjure make his texts challenging to realise. The poetry entices but the density, together with the increasing distance of the Cold War politics in the texts, leaves contemporary directors with clear decisions to make about how to adapt these open texts. In The New Dead: Medea Material, Henry works with some interesting imagery and conceptual territory. Lucinda Shaw as Medea, Guy Webster as Jason and Kimie Tsukakoshi as King Creon's daughter Glauce, the woman for whom Jason forsakes his wife Medea, each reference different aspects of contemporary culture. Medea is a bitter, drunken, satin-gowned diva with bite; Jason - first seen lounging in front of the television with a beer in an image reminiscent of Sarah Kane's in-yer-face characterisation of Hippolytus in Phaedra's Love - has something of the rock star about him; and Glauce is a roller-skating, karaoke-singing, pole-dancing young temptress. The production is given a contemporary tone, dominated by Medea's twisted love and loss, rather than by any commentary on her circumstances. Its strength is the aesthetic Henry creates, supported by live electro-pop music, a band stage that stands as a metaphor for Jason's sea voyage, and multimedia that inserts images of the story unfolding beyond these characters' speeches as sorts of subconscious flashes. While Tsukakoshi is engaging throughout, there are moments when Shaw and Webster's performances - particularly in the songs - are diminished by a lack of clarity. The result is a piece that, while slightly lacking in its realisation at times, undoubtedly flags Henry's facility as an emerging director and what she wants to bring to the Brisbane theatre scene.

A view from the edge of the forest : recent progress in understanding the relationships of the insect orders

Research over the last two decades has significantly increased our understanding of the evolutionary position of the insects among other arthropods, and the relationships among the insect Orders. Many of these insights have been established through increasingly sophisticated analyses of DNA sequence data from a limited number of genes. Recent results have established the relationships of the Holometabola, but relationships among the hemimetabolous orders have been more difficult to elucidate. A strong consensus on the relationships among the Palaeoptera (Ephemeroptera and Odonata) and their relationship to the Neoptera has not emerged with all three possible resolutions supported by different data sets. While polyneopteran relationships generally have resisted significant resolution, it is now clear that termites, Isoptera, are nested within the cockroaches, Blattodea. The newly discovered order Mantophasmatodea is difficult to place with the balance of studies favouring Grylloblattodea as sister-group. While some studies have found the paraneopteran orders (Hemiptera, Thysanoptera, Phthiraptera and Psocoptera) monophyletic, evidence suggests that parasitic lice (Phthiraptera) have evolved from groups within the book and bark lice (Psocoptera), and may represent parallel evolutions of parasitism within two major louse groups. Within Holometabola, it is now clear that Hymenoptera are the sister to the other orders, that, in turn are divided into two clades, the Neuropteroidea (Coleoptera, Neuroptera and relatives) and the Mecopterida (Trichoptera, Lepidoptera, Diptera and their relatives). The enigmatic order Strepsiptera, the twisted wing insects, have now been placed firmly near Coleoptera, rejecting their close relationship to Diptera that was proposed some 15years ago primarily based on ribosomal DNA data. Phylogenomic-scale analyses are just beginning to be focused on the relationships of the insect orders, and this is where we expect to see resolution of palaeopteran and polyneopteran relationships. Future research will benefit from greater coordination between intra and inter-ordinal analyses. This will maximise the opportunities for appropriate outgroup choice at the intraordinal level and provide the background knowledge for the interordinal analyses to span the maximum phylogenetic scope within groups.

Ultraviolet Photodissociation of the N-Methylpyridinium Ion: Action Spectroscopy and Product Characterization

The ultraviolet photodissociation of gas-phase N-methylpyridinium ions is studied at room temperature using laser photodissociation mass spectrometry and structurally diagnostic ion-molecule reaction kinetics. The C5H5N-CH3+ (m/z 94), C5H5N-CD3+ (m/z 97), and C5D5N-CH3+(m/z 99) isotopologues are investigated, and it is shown that the N-methylpyridinium ion photodissociates by the loss of methane in the 36 000 - 43 000 cm(-1) (280 - 230 nm) region. The dissociation likely occurs on the ground state surface following internal conversion from the SI state. For each isotopologue, by monitoring the photofragmentation yield as a function of photon wavenumber, a broad vibronically featured band is recorded with origin (0-0) transitions assigned at 38 130, 38 140 and 38 320 cm(-1) for C5H5N-CH3+ C5H5N-CD3+ and C5D5N-CH3+, respectively. With the aid of quantum chemical calculations (CASSCF(6,6)/aug-cc-pVDZ), most of the observed vibronic detail is assigned to two in-plane ring deformation modes. Finally, using ion-molecule reactions, the methane coproduct at m/z 78 is confirmed as a 2-pyridinylium ion.

A computational model for BMP movement in sea urchin embryos

Bone morphogen proteins (BMPs) are distributed along a dorsal-ventral (DV) gradient in many developing embryos. The spatial distribution of this signaling ligand is critical for correct DV axis specification. In various species, BMP expression is spatially localized, and BMP gradient formation relies on BMP transport, which in turn requires interactions with the extracellular proteins Short gastrulation/Chordin (Chd) and Twisted gastrulation (Tsg). These binding interactions promote BMP movement and concomitantly inhibit BMP signaling. The protease Tolloid (Tld) cleaves Chd, which releases BMP from the complex and permits it to bind the BMP receptor and signal. In sea urchin embryos, BMP is produced in the ventral ectoderm, but signals in the dorsal ectoderm. The transport of BMP from the ventral ectoderm to the dorsal ectoderm in sea urchin embryos is not understood. Therefore, using information from a series of experiments, we adapt the mathematical model of Mizutani et al. (2005) and embed it as the reaction part of a one-dimensional reaction–diffusion model. We use it to study aspects of this transport process in sea urchin embryos. We demonstrate that the receptor-bound BMP concentration exhibits dorsally centered peaks of the same type as those observed experimentally when the ternary transport complex (Chd-Tsg-BMP) forms relatively quickly and BMP receptor binding is relatively slow. Similarly, dorsally centered peaks are created when the diffusivities of BMP, Chd, and Chd-Tsg are relatively low and that of Chd-Tsg-BMP is relatively high, and the model dynamics also suggest that Tld is a principal regulator of the system. At the end of this paper, we briefly compare the observed dynamics in the sea urchin model to a version that applies to the fly embryo, and we find that the same conditions can account for BMP transport in the two types of embryos only if Tld levels are reduced in sea urchin compared to fly.

The antigen 43 structure reveals a molecular velcro-like mechanism of autotransporter-mediated bacterial clumping

Aggregation and biofilm formation are critical mechanisms for bacterial resistance to host immune factors and antibiotics. Autotransporter (AT) proteins, which represent the largest group of outer-membrane and secreted proteins in Gram-negative bacteria, contribute significantly to these phenotypes. Despite their abundance and role in bacterial pathogenesis, most AT proteins have not been structurally characterized, and there is a paucity of detailed information with regard to their mode of action. Here we report the structure–function relationships of Antigen 43 (Ag43a), a prototypic self-associating AT protein from uropathogenic Escherichia coli. The functional domain of Ag43a displays a twisted L-shaped β-helical structure firmly stabilized by a 3D hydrogen-bonded scaffold. Notably, the distinctive Ag43a L shape facilitates self-association and cell aggregation. Combining all our data, we define a molecular “Velcro-like” mechanism of AT-mediated bacterial clumping, which can be tailored to fit different bacterial lifestyles such as the formation of biofilms.

3-Phenylisoquinolin-1(2H)-one

The title compound, C15H11NO, consists of a planar isoquinolinone group to which a phenyl ring is attached in a twisted fashion [dihedral angle = 39.44 (4)degrees]. The crystal packing is dominated by intermolecular N-H center dot center dot center dot O and C-H center dot center dot center dot O hydrogen bonds which define centrosymmetric dimeric entitities.

Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

1-(2,4-Dichlorophenyl)-3-[4-(dimethylamino)-phenyl]prop-2-enone

In the title compound, C17H15Cl2NO, the dimethylaminophenyl group is close to coplanar with the central propenone group [dihedral angle =13.1 (1)degrees between the mean planes], while the dichlorophenyl group is twisted from the plane [dihedral angle = 64.0 (1)degrees].In the crystal, C-H center dot center dot center dot O and weak C-H center dot center dot center dot pi interactions are formed between molecules.

Measuring the traction profile on sportsfields: Equipment development and testing

Traction is recognised as an important component of the overall playability and safety of a sportsfield. It relates to the "grip", or footing, provided through an athlete's shoe when in contact with the surface, and is normally measured by the torque generated when a weighted studded disc apparatus is dropped onto the turf and twisted manually. This paper describes the development of an automated traction tester, which mechanises the dropping and twisting of the weighted studded disc. By standardising these operational stages, more repeatable and reliable results can be expected than from the original hand-operated design where positioning of the disc and speed of rotation are controlled manually and so can vary from one measurement to the next. As well as measuring the maximum torque reached during rotation of the studded disc, the automated traction tester generates a profile of torque showing changes over time and calculates the angle through which the studded disc moved before reaching maximum torque. These aspects are now covered by a utility patent (PAT/AU/2004270767). Use of the automated traction tester is illustrated by comparative data for a range of warm-season turfgrasses, by comparisons of traction under different surface conditions generated by wear on Cynodon dactylon cultivars, and by the effects of environment, management and playing patterns on traction across a multi-use sports stadium.

Bibulocystis gloriosa sp nov (Pucciniales) on Caesalpinia scortechinii in Queensland, with comments on Spumula caesalpiniae

A rust causing leaf spotting and distortion of twigs and branches of Caesalpinia scortechinii in Queensland is described as the new species Bibulocystis gloriosa. Uredinia and telia occur on spotted pinnules, and pycnia, aecial uredinia and telia on galled and twisted leaf rachides, twigs and branches. B. gloriosa is similar to Bibulocystis viennotii on Albizia granulosa in New Caledonia in having a macrocyclic life cycle with all spore states, and teliospores with two fertile cells and two cysts. It differs in having aecial urediniospores and urediniospores with uniformly thickened walls and several scattered germ pores, rather than the apically thickened walls and equatorial germ pores of B. viennotii. Teliospores in the two species are similar in size, but those of B. gloriosa have proportionally larger fertile cells and smaller cysts than in B. viennotii. To date, B. gloriosa is known from only two localities in south-eastern Queensland. Comparison with the type specimen of Spumula caesalpiniae on Caesalpinia nuga from Indonesia has shown that the two rusts are generically distinct.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Ethyl 4-(4-chlorophenyl)-6-methyl-2-thioxo-1,2,3,4-tetrahydropyrimidine-5-carboxylate

In the title compound, C14H15ClN2O2S, the tetrahydropyrimidine ring adopts a twisted boat conformation with the carbonyl group in an s-trans conformation with respect to the C C double bond of the six-membered tetrahydropyrimidine ring. The molecular conformation is determined by an intramolecular C-H center dot center dot center dot pi interaction. The crystal structure is further stabilized by intermolecular N-H center dot center dot center dot O molecular chains and centrosymmetric N-H center dot center dot center dot S dimers.