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.

Exploiting the 1,2,3-triazolium motif in anion-templated formation of a bromide-selective rotaxane host assembly

Stimulated by the efficacy of copper (I) catalysed Huisgen-type 1,3-dipolar cycloaddition of terminal alkynes and organic azides to generate 1,4-disubstituted 1,2,3-triazole derivatives, the importance of ‘click’ chemistry in the synthesis of organic and biological molecular systems is ever increasing.[1] The mild reaction conditions have also led to this reaction gaining favour in the construction of interlocked molecular architectures.[2-4] In the majority of cases however, the triazole group simply serves as a covalent linkage with no function in the resulting organic molecular framework. More recently a renewed interest has been shown in the transition metal coordination chemistry of triazole ligands.[3, 5, 6] In addition novel aryl macrocyclic and acyclic triazole based oligomers have been shown to recognise halide anions via cooperative triazole C5-H….anion hydrogen bonds.[7] In light of this it is surprising the potential anion binding affinity of the positively charged triazolium motif has not, with one notable exception,[8] been investigated. With the objective of manipulating the unique topological cavities of mechanically bonded molecules for anion recognition purposes, we have developed general methods of using anions to template the formation of interpenetrated and interlocked structures.[9-13] Herein we report the first examples of exploiting the 1,2,3-triazolium group in the anion templated formation of pseudorotaxane and rotaxane assemblies. In an unprecedented discovery the bromide anion is shown to be a superior templating reagent to chloride in the synthesis of a novel triazolium axle containing [2]rotaxane. Furthermore the resulting rotaxane interlocked host system exhibits the rare selectivity preference for bromide over chloride...

Alkyl radical geometry controls geminate cage recombination in alkylcobalamins

Abstract: The radical pair that results from photolysis of adenosylcob(II1)alamin (AdoCbl"') undergoes primary geminate recombination with a first-order rate constant of 1 x lo9 s-l. In contrast, methylcob(II1)alamin (CH3Cbl"') and aristeromicylcob(II1)alamin (AriCblII', the carbocyclic analogue of AdoCbl"' in which the ribofuranose ring oxygen has been replaced with a methylene group) does not undergo primary geminate recombination. The ribofwanose group enables a high rate of geminate recombination in the [Ado' Cbl"'] radical pair. This may be due to a stereoelectronic (p-anomeric) effect that maintains a pyramidal geometry at the 5'-carbon of the 5'-deoxyadenosyl radical, or it may be due to hindered rotation about the C4t-C5, bond such that /?-elimination to the olefin is prevented. Recombination in the geminate singlet radical pair is in competition with diffusive escape to form a solvent-separated radical pair. Hyperfine coupling from Co" promotes intersystem crossing to the triplet radical pair (Chagovetz, A. M.; Grissom, C. B. J. Am. Chem. SOC. 1993, 115, 12152). Recombination of the [CH3' Cbl"] radical pair is not prevented by a lack of intersystem crossing, as neither unlabeled or I3C-labeled CH3Cbl"' undergoes geminate recombination. There is only a small difference in the rate of diffusive recombination in the solvent cage for AdoCbl"', AriCbl"', and CH3Cbl"' following photolysis: 2.01 x 10" s-l, 2.20 x lo4 s-l, and 1.16 x lo4 s-l. The rate of diffusive recombination is limited by productive collisions and not by radical geometry or intersystem crossing. The CF3' radical that results from photolysis of (trifluoromethyl)cob(III)alamin (CF3Cbl"') maintains its pyramidal geometry and undergoes faster diffusive recombination in the solvent cage at 51 x lo4 s-l. The C-Co bond dissociation enthalpy in AriCbl"' is 37 f 1.4 kcaymol. The profound difference in geminate recombination rates for AdoCbl"' and CH3Cbl"' is consistent with their different biological roles as enzymatic cofactors: AdoCbl"' is an initiator of radical chain chemistry in the active site, whereas CH3Cbl"' is a methyl group donor in an S~2-type process.

Who's asking for what? Fundraising and leadership in Australian nonprofits

The nonprofit funding landscape is in flux. Many organisations are having to think differently and develop fresh skills either to enter the fundraising market or to cope better with rising competition for community and corporate support. This new reality affects boards, CEOs and fundraisers alike. Against this backdrop, our exploratory study aimed to build an evidence base and spark more discussion about: - the role Australian nonprofit CEOs and boards play in supporting fundraising/development; - current engagement levels; and - perceptions of leadership in fundraising from two possibly contrasting perspectives: NP leaders (board members and CEOs); and fundraisers. This research has been supported by the Perpetual Foundation, the EF and SL Gluyas Trust and the Edward Corbould Charitable Trust under the management of Perpetual Trustee Company Ltd.

Do sex and atopy influence cough outcome measurements in children?

Background Despite the commonality of cough and its burden, there are no published data on the relationship between atopy or sex on objectively measured cough frequency or subjective cough scores in children. In 202 children with and without cough, we determined the effect of sex and atopy on validated cough outcome measurements (cough receptor sensitivity [CRS], objective cough counts, and cough scores). We hypothesized that in contrast to adult data, sex does not influence cough outcome measures, and atopy is not a determinant of these cough measurements. Methods We combined data from four previous studies. Atopy (skin prick test), the concentration of capsaicin causing two and five or more coughs (C2 and C5, respectively), objectively measured cough frequency, and cough scores were determined and their relationship explored. The children’s (93 girls, 109 boys) mean age was 10.6 years (SD 2.9), and 56% had atopy. Results In multivariate analysis, CRS was influenced by age (C2 coefficient, 5.9; P = .034; C5 coefficient, 29.1; P = .0001). Atopy and sex did not significantly influence any of the cough outcomes (cough counts, C2, C5, cough score) in control subjects and children with cough. Conclusions Atopy does not influence important cough outcome measures in children with and without chronic cough. However, age, but not sex, influences CRS in children. Unlike adult data, sex does not affect objective counts or cough score in children with and without chronic cough. Studies on cough in children should be age matched, but matching for atopic status and sex is less important.

From bio-inspired computing to e-Biology

Several algorithms and techniques widely used in Computer Science have been adapted from, or inspired by, known biological phenomena. This is a consequence of the multidisciplinary background of most early computer scientists. The field has now matured, and permits development of tools and collaborative frameworks which play a vital role in advancing current biomedical research. In this paper, we briefly present examples of the former, and elaborate upon two of the latter, applied to immunological modelling and as a new paradigm in gene expression.

Association of variants in MMEL1 and CTLA4 with rheumatoid arthritis in the Han Chinese population

Background: The genome-wide association study era has made great progress in identifying susceptibility genes and genetic loci for rheumatoid arthritis (RA) in populations of White European ancestry. However, few studies have tried to dissect disease aetiopathogenesis in other ethnic populations. Objective: To investigate these associations in the Han Chinese population. Methods: Haplotypes from the HapMap database Chinese population were used to select tag-single-nucleotide polymorphisms (SNPs) (r2 =0.8) across 19 distinct RA genomic regions. A two phase case-control association study was performed, with 169 SNPs genotyped in phase I (n=571 cases, n=880 controls), and 64 SNPs achieving p<0.2 in the first phase being genotyped in phase II (n=464 cases, n=822 controls). Association statistics were calculated using permutation tests both unadjusted and adjusted for the number of markers studied. Results: Robust association was detected for MMEL1 and CTLA4 , and modest association was identified for another six loci: PADI4 , STAT4 , PRDM1 , CDK6 , TRAF1-C5 and KIF5A-PIP4K2C. All three markers genotyped in MMEL1 demonstrated association, with peak signal for rs3890745 (p=2.6×10 -5unadjusted, p=0.003 adjusted, OR=0.79). For CTLA4 , significance was detected for three of five variants showing association, with peak association for marker rs12992492 (p=4.3×10-5 unadjusted, p=0.0021 adjusted, OR=0.77). Lack of association of common variants in PTPN22 with RA in Han Chinese was confirmed. Conclusion: This study identifies MMEL1 and CTLA4 as RA susceptibility genes, provides suggestive evidence of association for a further six loci in the Han Chinese population and confirms lack of PTPN22 association in Asian populations. It also confirms the value of multiethnic population studies to help dissect disease aetiopathogenesis.