Aspects of elementary number theory illustrated in the spreadsheet environment

Number theory has in recent decades assumed a great practical importance, due primarily to its application to cryptography. This chapter discusses how elementary concepts of number theory may be illuminated and made accessible to upper secondary school students via appropriate spreadsheet models. In such environments, students can observe patterns, gain structural insight, form and test conjectures, and solve problems. The chapter begins by reviewing literature on the use of spreadsheets in general and the use of spreadsheets in number theory in particular. Two sample applications are then discussed. The first, factoring factorials, is presented and instructions are given to construct a model in Excel 2007. The second application, the RSA cryptosystem, is included because of its importance to Science, Technology, Engineering, and Mathematics (STEM) students. Number theoretic concepts relevant to RSA are discussed, and an outline of RSA. is given, with example. The chapter ends with instructions on how to construct a simple spreadsheet illustrating RSA.

Group law computations on Jacobians of hyperelliptic Curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form.

Planar difference functions

In 1980 Alltop produced a family of cubic phase sequences that nearly meet the Welch bound for maximum non-peak correlation magnitude. This family of sequences were shown by Wooters and Fields to be useful for quantum state tomography. Alltop’s construction used a function that is not planar, but whose difference function is planar. In this paper we show that Alltop type functions cannot exist in fields of characteristic 3 and that for a known class of planar functions, x^3 is the only Alltop type function.

Mutually unbiased bases as submodules and subspaces

Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with projective and affine planes. We propose a connection with higher dimensional projective geometries and projective Hjelmslev geometries. We show that this proposed geometric structure is present in several constructions of MUBs.

Fast formulas for computing cryptographic pairings

The most powerful known primitive in public-key cryptography is undoubtedly elliptic curve pairings. Upon their introduction just over ten years ago the computation of pairings was far too slow for them to be considered a practical option. This resulted in a vast amount of research from many mathematicians and computer scientists around the globe aiming to improve this computation speed. From the use of modern results in algebraic and arithmetic geometry to the application of foundational number theory that dates back to the days of Gauss and Euler, cryptographic pairings have since experienced a great deal of improvement. As a result, what was an extremely expensive computation that took several minutes is now a high-speed operation that takes less than a millisecond. This thesis presents a range of optimisations to the state-of-the-art in cryptographic pairing computation. Both through extending prior techniques, and introducing several novel ideas of our own, our work has contributed to recordbreaking pairing implementations.

A family of Alltop functions that are EA-inequivalent to the cubic function

Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (\emph{IEEE Trans. Inf. Theory} 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases (MUBs), a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called \emph{Alltop functions}. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.

Exploring the fundamental theorem of arithmetic in excel 2007

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

Spreadsheet Conditional Formatting Illuminates Investigations into Modular Arithmetic

Modular arithmetic has often been regarded as something of a mathematical curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digital security systems such as RSA, this important branch of mathematics is now considered essential knowledge for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular. This chapter describes how the modern graphical spreadsheet may be used to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Students may then gain structural insight and the foundations laid for applications to such areas as hashing, random number generation, and public-key cryptography.

Linking Theory and Practice: Performing a Reality Check on a Model of IS Success

The issue of ‘rigour vs. relevance’ in IS research has generated an intense, heated debate for over a decade. It is possible to identify, however, only a limited number of contributions on how to increase the relevance of IS research without compromising its rigour. Based on a lifecycle view of IS research, we propose the notion of ‘reality checks’ in order to review IS research outcomes in the light of actual industry demands. We assume that five barriers impact the efficient transfer of IS research outcomes; they are lack of awareness, lack of understandability, lack of relevance, lack of timeliness, and lack of applicability. In seeking to understand the effect of these barriers on the transfer of mature IS research into practice, we used focus groups. We chose DeLone and McLean’s IS success model as our stimulus because it is one of the more widely researched areas of IS.

A grounded theory study of nursing students’ experiences in the off-campus clinical setting

Poor workplace relations are an issue of concern in many workplaces and this phenomenon is not restricted to the nursing profession. The issue of workplace violence in nursing is well documented and there are an increasing number of studies which have investigated the notion of horizontal violence amongst graduate nurses. The impact that poor workplace relations has on the development of a professional identity by nursing students in the off-campus clinical setting is significant in light of the current global shortage of nurses. There is a dearth of knowledge in understanding how Australian undergraduate nursing students experience the off-campus clinical setting and subsequently develop a professional identity as a nurse. Therefore the aim of this study was to discover and describe the phenomena in order to develop a substantive theory that explains the experiences of the undergraduate nursing students in a regional setting. Constructivist grounded theory methods were utilised in the conduct of the study. A sample of 29 participants was recruited permitting the formulation of a substantive theory regarding the development of a professional identity in nursing students. This substantive theory contributes knowledge relevant to the undergraduate nursing students, nurse educators, nursing workforce planners, and the tertiary educational institutions offering nursing. This is achieved through discovering, describing and explaining the phenomenon of ‘anxiety’ which the nursing students experience as a result of the interrelationship and interactions of tradition bearing, staff and student performance. These interactions intersect to form expectations of where the student fits within the hierarchy of the facility and the nursing profession in general. An understanding of the issues associated with tradition bearing, staff performance, and student performance and the impact that the interaction of these conditions has upon the student’s developing professional identity as a nurse is necessary to allow for the implementation of corrective strategies.

Examination of social demographic status, disease status, health behaviours and quality of life in older Australians using the theory of planned behaviour model

The increase of life expectancy worldwide during the last three decades has increased age-related disability leading to the risk of loss of quality of life. How to improve quality of life including physical health and mental health for older people and optimize their life potential has become an important health issue. This study used the Theory of Planned Behaviour Model to examine factors influencing health behaviours, and the relationship with quality of life. A cross-sectional mailed survey of 1300 Australians over 50 years was conducted at the beginning of 2009, with 730 completed questionnaires returned (response rate 63%). Preliminary analysis reveals that physiological changes of old age, especially increasing waist circumference and co morbidity was closely related to health status, especially worse physical health summary score. Physical activity was the least adherent behaviour among the respondents compared to eating healthy food and taking medication regularly as prescribed. Increasing number of older people living alone with co morbidity of disease may be the barriers that influence their attitude and self control toward physical activity. A multidisciplinary and integrated approach including hospital and non hospital care is required to provide appropriate services and facilities toward older people.

Exposure to particle number, surface area and PM concentrations in pizzerias

The aim of this work was to quantify exposure to particles emitted by wood-fired ovens in pizzerias. Overall, 15 microenvironments were chosen and analyzed in a 14-month experimental campaign. Particle number concentration and distribution were measured simultaneously using a Condensation Particle Counter (CPC), a Scanning Mobility Particle Sizer (SMPS), an Aerodynamic Particle Sizer (APS). The surface area and mass distributions and concentrations, as well as the estimation of lung deposition surface area and PM1 were evaluated using the SMPS-APS system with dosimetric models, by taking into account the presence of aggregates on the basis of the Idealized Aggregate (IA) theory. The fraction of inhaled particles deposited in the respiratory system and different fractions of particulate matter were also measured by means of a Nanoparticle Surface Area Monitor (NSAM) and a photometer (DustTrak DRX), respectively. In this way, supplementary data were obtained during the monitoring of trends inside the pizzerias. We found that surface area and PM1 particle concentrations in pizzerias can be very high, especially when compared to other critical microenvironments, such as the transport hubs. During pizza cooking under normal ventilation conditions, concentrations were found up to 74, 70 and 23 times higher than background levels for number, surface area and PM1, respectively. A key parameter is the oven shape factor, defined as the ratio between the size of the face opening in respect

Tendering theory revisited

This paper discusses the content, origin and development of Tendering Theory as a theory of price determination. It demonstrates how tendering theory determines market prices and how it is different from game and decision theories, and that in the tendering process, with non-cooperative, simultaneous, single sealed bids with individual private valuations, extensive public information, a large number of bidders and a long sequence of tendering occasions, there develops a competitive equilibrium. The development of a competitive equilibrium means that the concept of the tender as the sum of a valuation and a strategy, which is at the core of tendering theory, cannot be supported and that there are serious empirical, theoretical and methodological inconsistencies in the theory.

Towards a relational theory of copyright law

This research provides a systematic and theoretical analysis of the digital challenges to the established exclusive regime of the economic rights enjoyed by authors (and related rightholders) under the law of copyright. Accordingly, this research has developed a relational theory of authorship and a relational approach to copyright, contending that the regulatory emphasis of copyright law should focus on the facilitation of the dynamic relations between the culture, the creators, the future creators, the users and the public, rather than the allocation of resources in a static world. In this networked digital world, the creative works and contents have become increasingly vital for people to engage in creativity and cultural innovation, and for the evolution of the economy. Hence, it is argued that today copyright owners, as content holders, have certain obligations to make their works accessible and available to the public under fair conditions. This research sets forward a number of recommendations for the reform of the current copyright system.

Information sharing with key suppliers : a transaction cost theory perspective

Purpose: The purpose of this paper is to explain variations in discretionary information shared between buyers and key suppliers. The paper also aims to examine how the extent of information shared affects buyers’ performance in terms of resource usage, output, and flexibility. ----- ----- Design/methodology/approach: The data for the paper comprise 221 Finnish and Swedish non-service companies obtained through a mail survey. The hypothesized relationships were tested using partial least squares modelling with reflective and formative constructs.----- ----- Findings: The results of the study suggest that (environmental and demand) uncertainty and interdependency can to some degree explain the extent of information shared between a buyer and key supplier. Furthermore, information sharing improves buyers’ performance with respect to resource usage, output, and flexibility.----- ----- Research limitations/implications: A limitation to the paper relates to the data, which only included buyers.Abetter approach would have been to collect data from both, buyers and key suppliers. Practical implications – Companies face a wide range of supply chain solutions that enable and encourage collaboration across organizations. This paper suggests a more selective and balanced approach toward adopting the solutions offered as the benefits are contingent on a number of factors such as uncertainty. Also, the risks of information sharing are far too high for a one size fits all approach.----- ----- Originality/value: The paper illustrates the applicability of transaction cost theory to the contemporary era of e-commerce. With this finding, transaction cost economics can provide a valuable lens with which to view and interpret interorganizational information sharing, a topic that has received much attention in the recent years.