Faster group operations on elliptic curves

This paper improves implementation techniques of Elliptic Curve Cryptography. We introduce new formulae and algorithms for the group law on Jacobi quartic, Jacobi intersection, Edwards, and Hessian curves. The proposed formulae and algorithms can save time in suitable point representations. To support our claims, a cost comparison is made with classic scalar multiplication algorithms using previous and current operation counts. Most notably, the best speeds are obtained from Jacobi quartic curves which provide the fastest timings for most scalar multiplication strategies benefiting from the proposed 12M + 5S + 1D point doubling and 7M + 3S + 1D point addition algorithms. Furthermore, the new addition algorithm provides an efficient way to protect against side channel attacks which are based on simple power analysis (SPA). Keywords: Efficient elliptic curve arithmetic,unified addition, side channel attack.

Elliptic curve algorithm integration in the secure shell transport layer

This document describes algorithms based on Elliptic Cryptography (ECC) for use within the Secure Shell (SSH) transport protocol. In particular, it specifies Elliptic Curve Diffie-Hellman (ECDH) key agreement, Elliptic Curve Menezes-Qu-Vanstone (ECMQV) key agreement, and Elliptic Curve Digital Signature Algorithm (ECDSA) for use in the SSH Transport Layer protocol.

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.

PIPP #1 - Pedestrian Interaction Patch Project

The Pedestrian Interaction Patch Project (PIPP) seeks to exert influence over and encourage abnormal pedestrian behavior. By placing an unadvertised (and non recording) interactive video manipulation system and projection source in a high traffic public area, the PIPP allows pedestrians to privately (and publically) re-engage with a previously inactive physical environment, like a commonly used walkway or corridor. This system, the results of which are projected in real time on the architectural surface, inadvertently provides pedestrians with questions around preconceived notions of self and space. In an attempt to re-activate our relationship with the physical surrounds we occupy each day the PIPP creates a new set of memories to be recalled as we re-enter known environments once PIPP has moved on and as such re-enlivens our relationship with the everyday architecture we stroll past everyday. The PIPP environment is controlled using the software program Isadora, devised by Mark Coniglio at Troika Ranch, and contains a series of video manipulation patches that are designed to not only grab the pedestrians attention but to also encourage a sense of play and interaction between the architecture, the digital environment, the initially unsuspecting participant(s) and the pedestrian audience. The PIPP was included as part of the planned walking tour for the “Playing in Urban Spaces” seminar day, and was an installation that ran for the length of the symposium in a reclaimed pedestrian space that was encountered by both the participants and general public during the course of the day long event. Ideally once discovered PIPP encouraged pedestrians to return through the course of the seminar day to see if the environmental patches had changed or altered, and changed their standard route to include the PIPP installation or to avoid it, either way, encouraging an active response to the pathways normally traveled or newly discovered each day.

PIPP #2 : Pedestrian Interaction Patch Project

Presented as part of the Sampled Festival at Sadlers Wells UK in January 2009, PIPP #2 continues the exploration of the first installation (PIPP #1 Leeds) and asks audiences to connect with an interactive work presented in the foyer of a major dance festival. Literally choreographing their own dances on the walls of the venue, pedestrians re-connect with the architectural surrounds generating unique memories of self.

Defending the patch : a labour geography analysis of unions’ construction of and responses to the problem of declining union membership amongst youth

If the trade union movement is to remain an influential force in the industrial, economic and socio/political arenas of industrialised nations it is vital that its recruitment of young members improve dramatically. Australian union membership levels have declined markedly over the last three decades and youth union membership levels have decreased more than any age group. Currently around 10% of young workers aged between 16-24 years are members of unions in Australia compared to 26% of workers aged 45-58 (Oliver, 2008). This decline has occurred throughout the union movement, in all states and in almost all industries and occupations. This research, which consists of interviews with union organisers and union officials, draws on perspectives from the labour geography literature to explore how union personnel located in various places, spaces and scales construct the issue of declining youth union membership. It explores the scale of connections within the labour movement and the extent to which these connections are leveraged to address the problem of youth union membership decline. To offer the reader a sense of context and perspective, the thesis firstly outlines the historical development of the union movement. It also reviews the literature on youth membership decline. Labour geography offers a rich and apposite analytical tool for investigation of this area. The notion of ‘scale’ as a dynamic, interactive, constructed and reconstructed entity (Ellem, 2006) is an appropriate lens for viewing youth-union membership issues. In this non-linear view, scale is a relational element which interplays with space, place and the environment (Howett, in Marston, 2000) rather than being ‘sequential’ and hierarchical. Importantly, the thesis investigates the notion of unions as ‘spaces of dependence’ (Cox, 1998a, p.2), organisations whose space is centred upon realising essential interests. It also considers the quality of unions’ interactions with others – their ‘spaces of engagement‘(Cox, 1998a, p.2), and the impact that this has upon their ability to recruit youth. The findings reveal that most respondents across the spectrum of the union movement attribute the decline in youth membership levels to factors external to the movement itself, such as changes to industrial relations legislation and the impact of globalisation on employment markets. However, participants also attribute responsibility for declining membership levels to the union movement itself, citing factors such as a lack of resourcing and a need to change unions’ perceived identity and methods of operation. The research further determined that networks of connections across the union movement are tenuous and, to date, are not being fully utilised to assist unions to overcome the youth recruitment dilemma. The study concludes that potential connections between unions are hampered by poor resourcing, workload issues and some deeply entrenched attitudes related to unions ‘defending (and maintaining) their patch’.

Interfacial debonding behavior of composite beam/plates with PZT patch

This paper studies interfacial debonding behavior of composite beams which include piezoelectric materials, adhesive and host beam. The focus is put on crack initiation and growth of the piezoelectric adhesive interface. Closed-form solutions of interface stresses and energy release rates are obtained for adhesive layer in the piezoelectric composite beams. Finite element analyses have been carried out to study the initiation and growth of interfaces crack for piezoelectric beams with interface element by ANSYS, in which the interface element of FE model is based on the cohesive zone models to characterize the fracture behavior of the interfacial debonding. The results have been compared with analystical solution, and the influence of different geometry and material parameters on the interfacial behavior of piezoelectric composite beams have been discussed.

Reduced-Size microstrip patch antenna for Bluetooth applications

A novel reduced-size microstrip rectangular patch antenna for Bluetooth operation is presented in this paper. The proposed antenna operates in the 2400 to 2484 MHz ISM Band. Although an air substrate is introduced, antenna occupies a small volume of 33.3×6.6×0.8 mm3. The gain and the impedance bandwidth of the antenna are predicted using a commercial Finite Element Method software package. The predicted results show good agreement with measured data.

Characteristics of a two-layer microstrip patch antenna for Bluetooth applications

A double-layer rectangular patch microstrip antenna suitable for Bluetooth applications is investigated. The patch is etched on a separate substrate which is suspended above the ground plane and supported by an MCX connector. The air gap between the patch and the ground plane increases the impedance bandwidth and can be used to tune the resonant frequency. This paper presents experimental results on the effects of various parameters on the antenna characteristics and provides guidelines for the design of such an antenna.