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.

Spectroscopy of selected copper group minerals : chalcophyllite and chenevixite-implications for hydrogen bonding

NIR and IR spectroscopy has been applied for detection of chemical species and the nature of hydrogen bonding in arsenate complexes. The structure and spectral properties of copper(II) arsenate minerals chalcophyllite and chenevixite are compared with copper(II) sulphate minerals devilline, chalcoalumite and caledonite. Split NIR bands in the electronic spectrum of two ranges 11700-8500 cm-1 and 8500-7200 cm-1 confirm distortion of octahedral symmetry for Cu(II) in the arsenate complexes. The observed bands with maxima at 9860 and 7750 cm-1 are assigned to Cu(II) transitions 2B1g ® 2B2g and 2B1g ® 2A1g. Overlapping bands in the NIR region 4500-4000 cm-1 is the effect of multi anions OH-, (AsO4)3- and (SO4)2-. The observation of broad and diffuse bands in the range 3700-2900 cm-1 confirms strong hydrogen bonding in chalcophyllite relative to chenevixite. The position of the water bending vibrations indicates the water is strongly hydrogen bonded in the mineral structure. The strong absorption feature centred at 1644 cm-1 in chalcophyllite indicates water is strongly hydrogen bonded in the mineral structure. The H2O-bending vibrations shift to low wavenumbers in chenevixite and an additional band observed at 1390 cm-1 is related to carbonate impurity. The characterisation of IR spectra by ν3 antisymmetric stretching vibrations of (SO4)2- and (AsO4)3 ions near 1100 and 800 cm-1 respectively is the result of isomorphic substitution for arsenate by sulphate in both the minerals of chalcophyllite and chenevixite.

Spectroscopy of selected copper group minerals : Ktenasite, orthoserpierite and Kipushite

The near-infrared (NIR) and infrared (IR) spectroscopy has been applied for characterisation of three complex Cu-Zn sulphate/phosphate minerals, namely ktenasite, orthoserpierite and kipushite. The spectral signatures of the three minerals are quite distinct in relation to their composition and structure. The effect of structural cations substitution (Zn2+ and Cu2+) on band shifts is significant both in the electronic and vibrational spectra of these Cu-Zn minerals. The variable Cu:Zn ratio between Zn-rich and Cu-rich compositions shows a strong effect on Cu(II) bands in the electronic spectra. The Cu(II) spectrum is most significant in kipushite (Cu-rich) with bands displayed at high wavenumbers at11390 and 7545 cm-1. The isomorphic substitution of Cu2+ for Zn2+ is reflected in the NIR and IR spectroscopic signatures. The multiple bands for 3 and 4 (SO4)2- stretching vibrations in ktenasite and orthoserpierite are attributed to the reduction of symmetry to the sulphate ion from Td to C2V. The IR spectrum of kipushite is characterised by strong (PO4)3- vibrational modes at 1090 and 990 cm-1. The range of IR absorption is higher in Ktenasite than in kipushite while it is intermediate in orthoserpierite.

Generic one round group key exchange in the standard model

Minimizing complexity of group key exchange (GKE) protocols is an important milestone towards their practical deployment. An interesting approach to achieve this goal is to simplify the design of GKE protocols by using generic building blocks. In this paper we investigate the possibility of founding GKE protocols based on a primitive called multi key encapsulation mechanism (mKEM) and describe advantages and limitations of this approach. In particular, we show how to design a one-round GKE protocol which satisfies the classical requirement of authenticated key exchange (AKE) security, yet without forward secrecy. As a result, we obtain the first one-round GKE protocol secure in the standard model. We also conduct our analysis using recent formal models that take into account both outsider and insider attacks as well as the notion of key compromise impersonation resilience (KCIR). In contrast to previous models we show how to model both outsider and insider KCIR within the definition of mutual authentication. Our analysis additionally implies that the insider security compiler by Katz and Shin from ACM CCS 2005 can be used to achieve more than what is shown in the original work, namely both outsider and insider KCIR.

Effect of poly(acrylic acid) end-group functionality on inhibition of calcium oxalate crystal growth

A number of series of poly(acrylic acids) (PAA) of differing end-groups and molecular weights prepared using atom transfer radical polymerization were used as inhibitors for the crystallization of calcium oxalate at 23 and 80°C. As measured by turbidimetry and conductivity and as expected from previous reports, all PAA series were most effective for inhibition of crystallization at molecular weights of 1500–4000. However, the extent of inhibition was in general strongly dependent on the hydrophobicity and molecular weight of the end-group. These results may be explicable in terms of adsorption/desorption of PAA to growth sites on crystallites. The overall effectiveness of the series didn't follow a simple trend with end-group hydrophobicity, suggesting self-assembly behavior or a balance between adsorption and desorption rates to crystallite surfaces may be critical in the mechanism of inhibition of calcium oxalate crystallization.

Effect of poly(acrylic acid) molecular mass and end-group functionality on calcium oxalate crystal morphology and growth

A number of series of poly(acrylic acids) (PAA) of differing end-groups and molecular mass were used to study the inhibition of calcium oxalate crystallization. The effects of the end-group on crystal speciation and morphology were significant and dramatic, with hexyl-isobutyrate end groups giving preferential formation of calcium oxalate dihydrate (COD) rather than the more stable calcium oxalate monohydrate (COM), while both more hydrophobic end-groups and less-hydrophobic end groups led predominantly to formation of the least thermodynamically stable form of calcium oxalate, calcium oxalate trihydrate. Conversely, molecular mass had little impact on calcium oxalate speciation or crystal morphology. It is probable that the observed effects are related to the rate of desorption of the PAA moiety from the crystal (lite) surfaces and that the results point to a major role for end-group as well as molecular mass in controlling desorption rate.

Crowd counting using group tracking and local features

In public venues, crowd size is a key indicator of crowd safety and stability. In this paper we propose a crowd counting algorithm that uses tracking and local features to count the number of people in each group as represented by a foreground blob segment, so that the total crowd estimate is the sum of the group sizes. Tracking is employed to improve the robustness of the estimate, by analysing the history of each group, including splitting and merging events. A simplified ground truth annotation strategy results in an approach with minimal setup requirements that is highly accurate.

The Queensland Cancer Physics Collaborative and the Quantitative Biomedical Imaging & Characterisation (Q-BIC) Research Group

a presentation about immersive visualised simulation systems, image analysis and GPGPU Techonology