Jacobi quartic curves revisited

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.

Clinical Use of the Volume-Time Curve for Endotracheal Tube Cuff Management

BACKGROUND: Previous investigation showed that the volume-time curve technique could be an alternative for endotracheal tube (ETT) cuff management. However, the clinical impact of the volume-time curve application has not been documented. the purpose of this study was to compare the occurrence and intensity of a sore throat, cough, thoracic pain, and pulmonary function between these 2 techniques for ETT cuff management: volume-time curve technique versus minimal occlusive volume (MOV) technique after coronary artery bypass grafting. METHODS: A total of 450 subjects were randomized into 2 groups for cuff management after intubation: MOV group (n = 222) and volume-time curve group (n = 228). We measured cuff pressure before extubation. We performed spirometry 24 h before and after surgery. We graded sore throat and cough according to a 4-point scale at 1, 24, 72, and 120 h after extubation and assessed thoracic pain at 24 h after extubation and quantified the level of pain by a 10-point scale. RESULTS: the volume-time curve group presented significantly lower cuff pressure (30.9 +/- 2.8 vs 37.7 +/- 3.4 cm H2O), less incidence and intensity of sore throat (1 h, 23.7 vs 51.4%; and 24 h, 18.9 vs 40.5%, P < .001), cough (1 h, 19.3 vs 48.6%; and 24 h, 18.4 vs 42.3%, P < .001), thoracic pain (5.2 +/- 1.8 vs 7.1 +/- 1.7), better preservation of FVC (49.5 +/- 9.9 vs 41.8 +/- 12.9%, P = .005), and FEV1, (46.6 +/- 1.8 vs 38.6 +/- 1.4%, P = .005) compared with the MOV group. CONCLUSIONS: the subjects who received the volume-time curve technique for ETT cuff management presented a significantly lower incidence and severity of sore throat and cough, less thoracic pain, and minimally impaired pulmonary function than those subjects who received the MOV technique during the first 24 h after coronary artery bypass grafting.

Antimicrobial prescribing in Nursing Homes in Northern Ireland: Results of Two Point-Prevalence Surveys

BACKGROUND: In 2005, the European Commission recommended that all member states should establish or strengthen surveillance systems for monitoring the use of antimicrobial agents. There is no evidence in the literature of any surveillance studies having been specifically conducted in nursing homes (NHs) in Northern Ireland (NI).

OBJECTIVE: The aim of this study was to determine the prevalence of antimicrobial prescribing and its relationship with certain factors (e.g. indwelling urinary catheterization, urinary incontinence, disorientation, etc.) in NH residents in NI.

METHODS: This project was carried out in NI as part of a wider European study under the protocols of the European Surveillance of Antimicrobial Consumption group. Two point-prevalence surveys (PPSs) were conducted in 30 NHs in April and November 2009. Data were obtained from nursing notes, medication administration records and staff in relation to antimicrobial prescribing, facility and resident characteristics and were analysed descriptively.

RESULTS: The point prevalence of antimicrobial prescribing was 13.2% in April 2009 and 10.7% in November 2009, with a 10-fold difference existing between the NHs with the highest and lowest antimicrobial prescribing prevalence during both PPSs. The same NH had the highest rate of antimicrobial prescribing during both April (30.6%) and November (26.0%). The group of antimicrobials most commonly prescribed was the penicillins (April 28.6%, November 27.5%) whilst the most prevalent individual antimicrobial prescribed was trimethoprim (April 21.3%, November 24.3%). The majority of antimicrobials were prescribed for the purpose of preventing urinary tract infections (UTIs) in both April (37.8%) and in November (46.7%), with 5% of all participating residents being prescribed an antimicrobial for this reason. Some (20%) antimicrobials were prescribed at inappropriate doses, particularly those which were used for the purpose of preventing UTIs. Indwelling urinary catheterization and wounds were significant risk factors for antimicrobial use in April [odds ratio {OR} (95% CI) 2.0 (1.1, 3.5) and 1.8 (1.1, 3.0), respectively] but not in November 2009 [OR (95% CI) 1.6 (0.8, 3.2) and 1.2 (0.7, 2.2), respectively]. Other resident factors, e.g. disorientation, immobility and incontinence, were not associated with antimicrobial use. Furthermore, none of the NH characteristics investigated (e.g. number of beds, hospitalization episodes, number of general practitioners, etc.) were found to be associated with antimicrobial use in either April or November 2009.

CONCLUSIONS: This study has identified a high overall rate of antimicrobial use in NHs in NI, with variability evident both within and between homes. More research is needed to understand which factors influence antimicrobial use and to determine the appropriateness of antimicrobial prescribing in this population in general and more specifically in the management of recurrent UTIs.

Does point of care prothrombin time measurement reduce the transfusion of fresh frozen plasma in patients undergoing major surgery? The POC-OP randomized-controlled trial

BACKGROUND: Bleeding is a frequent complication during surgery. The intraoperative administration of blood products, including packed red blood cells, platelets and fresh frozen plasma (FFP), is often live saving. Complications of blood transfusions contribute considerably to perioperative costs and blood product resources are limited. Consequently, strategies to optimize the decision to transfuse are needed. Bleeding during surgery is a dynamic process and may result in major blood loss and coagulopathy due to dilution and consumption. The indication for transfusion should be based on reliable coagulation studies. While hemoglobin levels and platelet counts are available within 15 minutes, standard coagulation studies require one hour. Therefore, the decision to administer FFP has to be made in the absence of any data. Point of care testing of prothrombin time ensures that one major parameter of coagulation is available in the operation theatre within minutes. It is fast, easy to perform, inexpensive and may enable physicians to rationally determine the need for FFP. METHODS/DESIGN: The objective of the POC-OP trial is to determine the effectiveness of point of care prothrombin time testing to reduce the administration of FFP. It is a patient and assessor blind, single center randomized controlled parallel group trial in 220 patients aged between 18 and 90 years undergoing major surgery (any type, except cardiac surgery and liver transplantation) with an estimated blood loss during surgery exceeding 20% of the calculated total blood volume or a requirement of FFP according to the judgment of the physicians in charge. Patients are randomized to usual care plus point of care prothrombin time testing or usual care alone without point of care testing. The primary outcome is the relative risk to receive any FFP perioperatively. The inclusion of 110 patients per group will yield more than 80% power to detect a clinically relevant relative risk of 0.60 to receive FFP of the experimental as compared with the control group. DISCUSSION: Point of care prothrombin time testing in the operation theatre may reduce the administration of FFP considerably, which in turn may decrease costs and complications usually associated with the administration of blood products. TRIAL REGISTRATION: NCT00656396.

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.

The effect of exercise training on sleep quality in heart failure

Background: Heart failure is a serious condition estimated to affect 1.5-2.0% of the Australian population with a point prevalence of approximately 1% in people aged 50-59 years, 10% in people aged 65 years or more and over 50% in people aged 85 years or over (National Heart Foundation of Australian and the Cardiac Society of Australia and New Zealand, 2006). Sleep disturbances are a common complaint of persons with heart failure. Disturbances of sleep can worsen heart failure symptoms, impair independence, reduce quality of life and lead to increased health care utilisation in patients with heart failure. Previous studies have identified exercise as a possible treatment for poor sleep in patients without cardiac disease however there is limited evidence of the effect of this form of treatment in heart failure. Aim: The primary objective of this study was to examine the effect of a supervised, hospital-based exercise training programme on subjective sleep quality in heart failure patients. Secondary objectives were to examine the association between changes in sleep quality and changes in depression, exercise performance and body mass index. Methods: The sample for the study was recruited from metropolitan and regional heart failure services across Brisbane, Queensland. Patients with a recent heart failure related hospital admission who met study inclusion criteria were recruited. Participants were screened by specialist heart failure exercise staff at each site to ensure exercise safety prior to study enrolment. Demographic data, medical history, medications, Pittsburgh Sleep Quality Index score, Geriatric Depression Score, exercise performance (six minute walk test), weight and height were collected at Baseline. Pittsburgh Sleep Quality Index score, Geriatric Depression Score, exercise performance and weight were repeated at 3 months. One hundred and six patients admitted to hospital with heart failure were randomly allocated to a 3-month disease-based management programme of education and self-management support including standard exercise advice (Control) or to the same disease management programme as the Control group with the addition of a tailored physical activity program (Intervention). The intervention consisted of 1 hour of aerobic and resistance exercise twice a week. Programs were designed and supervised by an exercise specialist. The main outcome measure was achievement of a clinically significant change (.3 points) in global Pittsburgh Sleep Quality score. Results: Intervention group participants reported significantly greater clinical improvement in global sleep quality than Control (p=0.016). These patients also exhibited significant improvements in component sleep disturbance (p=0.004), component sleep quality (p=0.015) and global sleep quality (p=0.032) after 3 months of supervised exercise intervention. Improvements in sleep quality correlated with improvements in depression (p<0.001) and six minute walk distance (p=0.04). When study results were examined categorically, with subjects classified as either "poor" or "good" sleepers, subjects in the Control group were significantly more likely to report "poor" sleep at 3 months (p=0.039) while Intervention participants were likely to report "good" sleep at this time (p=0.08). Conclusion: Three months of supervised, hospital based, aerobic and resistance exercise training improved subjective sleep quality in patients with heart failure. This is the first randomised controlled trial to examine the role of aerobic and resistance exercise training in the improvement of sleep quality for patients with this disease. While this study establishes exercise as a therapy for poor sleep quality, further research is needed to investigate the effect of exercise training on objective parameters of sleep in this population.

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.

Prevention of type 2 diabetes with lifestyle intervention - emphasis on dietary composition and identification of high-risk individuals

Type 2 diabetes is an increasing, serious, and costly public health problem. The increase in the prevalence of the disease can mainly be attributed to changing lifestyles leading to physical inactivity, overweight, and obesity. These lifestyle-related risk factors offer also a possibility for preventive interventions. Until recently, proper evidence regarding the prevention of type 2 diabetes has been virtually missing. To be cost-effective, intensive interventions to prevent type 2 diabetes should be directed to people at an increased risk of the disease. The aim of this series of studies was to investigate whether type 2 diabetes can be prevented by lifestyle intervention in high-risk individuals, and to develop a practical method to identify individuals who are at high risk of type 2 diabetes and would benefit from such an intervention. To study the effect of lifestyle intervention on diabetes risk, we recruited 522 volunteer, middle-aged (aged 40 - 64 at baseline), overweight (body mass index > 25 kg/m2) men (n = 172) and women (n = 350) with impaired glucose tolerance to the Diabetes Prevention Study (DPS). The participants were randomly allocated either to the intensive lifestyle intervention group or the control group. The control group received general dietary and exercise advice at baseline, and had annual physician's examination. The participants in the intervention group received, in addition, individualised dietary counselling by a nutritionist. They were also offered circuit-type resistance training sessions and were advised to increase overall physical activity. The intervention goals were to reduce body weight (5% or more reduction from baseline weight), limit dietary fat (< 30% of total energy consumed) and saturated fat (< 10% of total energy consumed), and to increase dietary fibre intake (15 g / 1000 kcal or more) and physical activity (≥ 30 minutes/day). Diabetes status was assessed annually by a repeated 75 g oral glucose tolerance testing. First analysis on end-points was completed after a mean follow-up of 3.2 years, and the intervention phase was terminated after a mean duration of 3.9 years. After that, the study participants continued to visit the study clinics for the annual examinations, for a mean of 3 years. The intervention group showed significantly greater improvement in each intervention goal. After 1 and 3 years, mean weight reductions were 4.5 and 3.5 kg in the intervention group and 1.0 kg and 0.9 kg in the control group. Cardiovascular risk factors improved more in the intervention group. After a mean follow-up of 3.2 years, the risk of diabetes was reduced by 58% in the intervention group compared with the control group. The reduction in the incidence of diabetes was directly associated with achieved lifestyle goals. Furthermore, those who consumed moderate-fat, high-fibre diet achieved the largest weight reduction and, even after adjustment for weight reduction, the lowest diabetes risk during the intervention period. After discontinuation of the counselling, the differences in lifestyle variables between the groups still remained favourable for the intervention group. During the post-intervention follow-up period of 3 years, the risk of diabetes was still 36% lower among the former intervention group participants, compared with the former control group participants. To develop a simple screening tool to identify individuals who are at high risk of type 2 diabetes, follow-up data of two population-based cohorts of 35-64 year old men and women was used. The National FINRISK Study 1987 cohort (model development data) included 4435 subjects, with 182 new drug-treated cases of diabetes identified during ten years, and the FINRISK Study 1992 cohort (model validation data) included 4615 subjects, with 67 new cases of drug-treated diabetes during five years, ascertained using the Social Insurance Institution's Drug register. Baseline age, body mass index, waist circumference, history of antihypertensive drug treatment and high blood glucose, physical activity and daily consumption of fruits, berries or vegetables were selected into the risk score as categorical variables. In the 1987 cohort the optimal cut-off point of the risk score identified 78% of those who got diabetes during the follow-up (= sensitivity of the test) and 77% of those who remained free of diabetes (= specificity of the test). In the 1992 cohort the risk score performed equally well. The final Finnish Diabetes Risk Score (FINDRISC) form includes, in addition to the predictors of the model, a question about family history of diabetes and the age category of over 64 years. When applied to the DPS population, the baseline FINDRISC value was associated with diabetes risk among the control group participants only, indicating that the intensive lifestyle intervention given to the intervention group participants abolished the diabetes risk associated with baseline risk factors. In conclusion, the intensive lifestyle intervention produced long-term beneficial changes in diet, physical activity, body weight, and cardiovascular risk factors, and reduced diabetes risk. Furthermore, the effects of the intervention were sustained after the intervention was discontinued. The FINDRISC proved to be a simple, fast, inexpensive, non-invasive, and reliable tool to identify individuals at high risk of type 2 diabetes. The use of FINDRISC to identify high-risk subjects, followed by lifestyle intervention, provides a feasible scheme in preventing type 2 diabetes, which could be implemented in the primary health care system.

Capturability analysis of a geometric guidance law in relative velocity space

In this paper, a relative velocity approach is used to analyze the capturability of a geometric guidance law. Point mass models are assumed for both the missile and the target. The speeds of the missile and target are assumed to remain constant throughout the engagement. Lateral acceleration, obtained from the guidance law, is applied to change the path of the missile. The kinematic equations for engagements in the horizontal plane are derived in the relative velocity space. Some analytical results for the capture region are obtained for non-maneuvering and maneuvering targets. For non-maneuvering targets it is enough for the navigation gain to be a constant to intercept the target, while for maneuvering targets a time varying navigation gain is needed for interception. These results are then verified through numerical simulations.

Fault Attacks on Pairing-Based Protocols Revisited

Several papers have studied fault attacks on computing a pairing value e(P, Q), where P is a public point and Q is a secret point. In this paper, we observe that these attacks are in fact effective only on a small number of pairing-based protocols, and that too only when the protocols are implemented with specific symmetric pairings. We demonstrate the effectiveness of the fault attacks on a public-key encryption scheme, an identity-based encryption scheme, and an oblivious transfer protocol when implemented with a symmetric pairing derived from a supersingular elliptic curve with embedding degree 2.

Avaliação in vitro do efeito da incorporação de clorexidina nas propriedades físicas e na atividade antibacteriana de resinas acrílicas autopolimerizáveis

Objetivo. O objetivo do presente estudo foi avaliar o efeito da incorporação de diacetato de clorexidina (CDA), em diferentes concentrações e tempos de armazenamento, nas propriedades físicas e na atividade antibacteriana de resinas acrílicas, utilizadas na confecção de coroas e pontes provisórias. Métodos. Fase I: Foram confeccionados 150 corpos de prova retangulares (3,0 mm X 10 mm X 64 mm), de acordo com a norma ISO 1567 e 150 corpos de prova quadrados (10 mm X 10 mm X 2,0 mm), utilizando-se duas resinas acrílicas autopolimerizáveis, Duralay (Reliance Dental Mfg. Co.) e Dencor (Clássico). Os corpos de prova foram distribuídos em 30 grupos (n=10/grupo) de acordo com a concentração de CDA incorporada às resinas (p/p) (A) 0%, (B) 1%, (C) 2%, (D) 4%, (E) 5%, em função do tempo de armazenamento em água destilada, a 37C (T0 2h, T1 7 dias, T2 30 dias). Foram realizados os ensaios de microdureza Knoop, em microdurômetro Micromet 5104, Buehler (N), rugosidade superficial (Ra), em rugosímetro digital Mitutoyo Surftest SJ-201 (n=5) e resistência à flexão em três pontos (MPa), em uma máquina de ensaio universal EMIC MF 200 DL (n=5). Fase II: Adicionalmente, a atividade antibacteriana dos materiais sobre Streptococcus mutans foi determinada através da realização de testes de difusão em meio BHI, sendo para isso confeccionados 30 corpos de prova em forma de disco (12 mm X 3,0 mm) com as mesmas 5 concentrações (n=3/grupo). Os resultados foram tabulados e submetidos à análise estatística three-way ANOVA (Fase I) e two-way ANOVA (Fase II). Resultados. ANOVA mostrou que a adição de CDA não provocou alteração significativa na resistência à flexão dos materiais testados. A resistência à flexão é inversamente proporcional ao tempo para a resina Dencor e diretamente proporcional ao tempo para a resina Duralay. Houve aumento da microdureza com o acréscimo de CDA ao material Dencor com relação ao grupo controle, enquanto que no material Duralay a CDA não interferiu significativamente nesta propriedade. A rugosidade superficial aumentou significativamente (p<0,001) com o tempo e com o aumento da concentração de clorexidina na resina Dencor e não provocou alteração significativa em Duralay. Os testes de difusão em ágar demonstraram atividade antimicrobiana significativa (p<0,05) em todos os grupos, quando comparados ao grupo-controle. A inibição ao crescimento de Streptococcus mutans foi maior com o aumento da concentração desta substância. A resina Dencor apresentou maior halo de inibição do que a resina Duralay. Conclusões. Os resultados deste estudo sugerem que a incorporação de clorexidina aos materiais testados exibiu efeito antibacteriano contra S. mutans, sem contudo afetar de maneira crítica as propriedades físicas avaliadas.

Hardware design of cryptographic accelerators

With the rapid growth of the Internet and digital communications, the volume of sensitive electronic transactions being transferred and stored over and on insecure media has increased dramatically in recent years. The growing demand for cryptographic systems to secure this data, across a multitude of platforms, ranging from large servers to small mobile devices and smart cards, has necessitated research into low cost, flexible and secure solutions. As constraints on architectures such as area, speed and power become key factors in choosing a cryptosystem, methods for speeding up the development and evaluation process are necessary. This thesis investigates flexible hardware architectures for the main components of a cryptographic system. Dedicated hardware accelerators can provide significant performance improvements when compared to implementations on general purpose processors. Each of the designs proposed are analysed in terms of speed, area, power, energy and efficiency. Field Programmable Gate Arrays (FPGAs) are chosen as the development platform due to their fast development time and reconfigurable nature. Firstly, a reconfigurable architecture for performing elliptic curve point scalar multiplication on an FPGA is presented. Elliptic curve cryptography is one such method to secure data, offering similar security levels to traditional systems, such as RSA, but with smaller key sizes, translating into lower memory and bandwidth requirements. The architecture is implemented using different underlying algorithms and coordinates for dedicated Double-and-Add algorithms, twisted Edwards algorithms and SPA secure algorithms, and its power consumption and energy on an FPGA measured. Hardware implementation results for these new algorithms are compared against their software counterparts and the best choices for minimum area-time and area-energy circuits are then identified and examined for larger key and field sizes. Secondly, implementation methods for another component of a cryptographic system, namely hash functions, developed in the recently concluded SHA-3 hash competition are presented. Various designs from the three rounds of the NIST run competition are implemented on FPGA along with an interface to allow fair comparison of the different hash functions when operating in a standardised and constrained environment. Different methods of implementation for the designs and their subsequent performance is examined in terms of throughput, area and energy costs using various constraint metrics. Comparing many different implementation methods and algorithms is nontrivial. Another aim of this thesis is the development of generic interfaces used both to reduce implementation and test time and also to enable fair baseline comparisons of different algorithms when operating in a standardised and constrained environment. Finally, a hardware-software co-design cryptographic architecture is presented. This architecture is capable of supporting multiple types of cryptographic algorithms and is described through an application for performing public key cryptography, namely the Elliptic Curve Digital Signature Algorithm (ECDSA). This architecture makes use of the elliptic curve architecture and the hash functions described previously. These components, along with a random number generator, provide hardware acceleration for a Microblaze based cryptographic system. The trade-off in terms of performance for flexibility is discussed using dedicated software, and hardware-software co-design implementations of the elliptic curve point scalar multiplication block. Results are then presented in terms of the overall cryptographic system.

Algebraic Geometric Codes and their relation to Cryptography using Elliptic Curves

Communication is the process of transmitting data across channel. Whenever data is transmitted across a channel, errors are likely to occur. Coding theory is a stream of science that deals with finding efficient ways to encode and decode data, so that any likely errors can be detected and corrected. There are many methods to achieve coding and decoding. One among them is Algebraic Geometric Codes that can be constructed from curves. Cryptography is the science ol‘ security of transmitting messages from a sender to a receiver. The objective is to encrypt message in such a way that an eavesdropper would not be able to read it. A eryptosystem is a set of algorithms for encrypting and decrypting for the purpose of the process of encryption and decryption. Public key eryptosystem such as RSA and DSS are traditionally being prel‘en‘ec| for the purpose of secure communication through the channel. llowever Elliptic Curve eryptosystem have become a viable altemative since they provide greater security and also because of their usage of key of smaller length compared to other existing crypto systems. Elliptic curve cryptography is based on group of points on an elliptic curve over a finite field. This thesis deals with Algebraic Geometric codes and their relation to Cryptography using elliptic curves. Here Goppa codes are used and the curves used are elliptic curve over a finite field. We are relating Algebraic Geometric code to Cryptography by developing a cryptographic algorithm, which includes the process of encryption and decryption of messages. We are making use of fundamental properties of Elliptic curve cryptography for generating the algorithm and is used here to relate both.

Mersenne numbers

These notes have been issued on a small scale in 1983 and 1987 and on request at other times. This issue follows two items of news. First, WaIter Colquitt and Luther Welsh found the 'missed' Mersenne prime M110503 and advanced the frontier of complete Mp-testing to 139,267. In so doing, they terminated Slowinski's significant string of four consecutive Mersenne primes. Secondly, a team of five established a non-Mersenne number as the largest known prime. This result terminated the 1952-89 reign of Mersenne primes. All the original Mersenne numbers with p < 258 were factorised some time ago. The Sandia Laboratories team of Davis, Holdridge & Simmons with some little assistance from a CRAY machine cracked M211 in 1983 and M251 in 1984. They contributed their results to the 'Cunningham Project', care of Sam Wagstaff. That project is now moving apace thanks to developments in technology, factorisation and primality testing. New levels of computer power and new computer architectures motivated by the open-ended promise of parallelism are now available. Once again, the suppliers may be offering free buildings with the computer. However, the Sandia '84 CRAY-l implementation of the quadratic-sieve method is now outpowered by the number-field sieve technique. This is deployed on either purpose-built hardware or large syndicates, even distributed world-wide, of collaborating standard processors. New factorisation techniques of both special and general applicability have been defined and deployed. The elliptic-curve method finds large factors with helpful properties while the number-field sieve approach is breaking down composites with over one hundred digits. The material is updated on an occasional basis to follow the latest developments in primality-testing large Mp and factorising smaller Mp; all dates derive from the published literature or referenced private communications. Minor corrections, additions and changes merely advance the issue number after the decimal point. The reader is invited to report any errors and omissions that have escaped the proof-reading, to answer the unresolved questions noted and to suggest additional material associated with this subject.

Transcendental Brauer groups of products of CM elliptic curves

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O_K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface ExE. The results for the odd order torsion also apply to the Brauer group of the K3 surface Kum(ExE). We describe explicitly the elliptic curves E/Q with complex multiplication by O_K such that the Brauer group of ExE contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(ExE), while there is no obstruction coming from the algebraic part of the Brauer group.