Converse results to the Wiener attack on RSA

A well-known attack on RSA with low secret-exponent d was given by Wiener about 15 years ago. Wiener showed that using continued fractions, one can efficiently recover the secret-exponent d from the public key (N,e) as long as d < N 1/4. Interestingly, Wiener stated that his attack may sometimes also work when d is slightly larger than N 1/4. This raises the question of how much larger d can be: could the attack work with non-negligible probability for d=N 1/4 + ρ for some constant ρ > 0? We answer this question in the negative by proving a converse to Wiener’s result. Our result shows that, for any fixed ε > 0 and all sufficiently large modulus lengths, Wiener’s attack succeeds with negligible probability over a random choice of d < N δ (in an interval of size Ω(N δ )) as soon as δ > 1/4 + ε. Thus Wiener’s success bound dWiener attack, which are guaranteed to succeed even when δ > 1/4. The known attacks in this class (by Verheul and Van Tilborg and Dujella) run in exponential time, so it is natural to ask whether there exists an attack in this class with subexponential run-time. Our second converse result answers this question also in the negative.

A natural model for architecture : the nature of the evolutionary model 1995

John Frazer's architectural work is inspired by living and generative processes. Both evolutionary and revolutionary, it explores informatin ecologies and the dynamics of the spaces between objects. Fuelled by an interest in the cybernetic work of Gordon Pask and Norbert Wiener, and the possibilities of the computer and the "new science" it has facilitated, Frazer and his team of collaborators have conducted a series of experiments that utilize genetic algorithms, cellular automata, emergent behaviour, complexity and feedback loops to create a truly dynamic architecture. Frazer studied at the Architectural Association (AA) in London from 1963 to 1969, and later became unit master of Diploma Unit 11 there. He was subsequently Director of Computer-Aided Design at the University of Ulter - a post he held while writing An Evolutionary Architecture in 1995 - and a lecturer at the University of Cambridge. In 1983 he co-founded Autographics Software Ltd, which pioneered microprocessor graphics. Frazer was awarded a person chair at the University of Ulster in 1984. In Frazer's hands, architecture becomes machine-readable, formally open-ended and responsive. His work as computer consultant to Cedric Price's Generator Project of 1976 (see P84)led to the development of a series of tools and processes; these have resulted in projects such as the Calbuild Kit (1985) and the Universal Constructor (1990). These subsequent computer-orientated architectural machines are makers of architectural form beyond the full control of the architect-programmer. Frazer makes much reference to the multi-celled relationships found in nature, and their ongoing morphosis in response to continually changing contextual criteria. He defines the elements that describe his evolutionary architectural model thus: "A genetic code script, rules for the development of the code, mapping of the code to a virtual model, the nature of the environment for the development of the model and, most importantly, the criteria for selection. In setting out these parameters for designing evolutionary architectures, Frazer goes beyond the usual notions of architectural beauty and aesthetics. Nevertheless his work is not without an aesthetic: some pieces are a frenzy of mad wire, while others have a modularity that is reminiscent of biological form. Algorithms form the basis of Frazer's designs. These algorithms determine a variety of formal results dependent on the nature of the information they are given. His work, therefore, is always dynamic, always evolving and always different. Designing with algorithms is also critical to other architects featured in this book, such as Marcos Novak (see p150). Frazer has made an unparalleled contribution to defining architectural possibilities for the twenty-first century, and remains an inspiration to architects seeking to create responsive environments. Architects were initially slow to pick up on the opportunities that the computer provides. These opportunities are both representational and spatial: computers can help architects draw buildings and, more importantly, they can help architects create varied spaces, both virtual and actual. Frazer's work was groundbreaking in this respect, and well before its time.

Contracting bubbles in Hele-Shaw cells with a power-law fluid

The problem of bubble contraction in a Hele-Shaw cell is studied for the case in which the surrounding fluid is of power-law type. A small perturbation of the radially symmetric problem is first considered, focussing on the behaviour just before the bubble vanishes, it being found that for shear-thinning fluids the radially symmetric solution is stable, while for shear-thickening fluids the aspect ratio of the bubble boundary increases. The borderline (Newtonian) case considered previously is neutrally stable, the bubble boundary becoming elliptic in shape with the eccentricity of the ellipse depending on the initial data. Further light is shed on the bubble contraction problem by considering a long thin Hele-Shaw cell: for early times the leading-order behaviour is one-dimensional in this limit; however, as the bubble contracts its evolution is ultimately determined by the solution of a Wiener-Hopf problem, the transition between the long-thin limit and the extinction limit in which the bubble vanishes being described by what is in effect a similarity solution of the second kind. This same solution describes the generic (slit-like) extinction behaviour for shear-thickening fluids, the interface profiles that generalise the ellipses that characterise the Newtonian case being constructed by the Wiener-Hopf calculation.

Linear stern waves in finite depth channels

This paper formulates an analytically tractable problem for the wake generated by a long flat bottom ship by considering the steady free surface flow of an inviscid, incompressible fluid emerging from beneath a semi-infinite rigid plate. The flow is considered to be irrotational and two-dimensional so that classical potential flow methods can be exploited. In addition, it is assumed that the draft of the plate is small compared to the depth of the channel. The linearised problem is solved exactly using a Fourier transform and the Wiener-Hopf technique, and it is shown that there is a family of subcritical solutions characterised by a train of sinusoidal waves on the downstream free surface. The amplitude of these waves decreases as the Froude number increases. Supercritical solutions are also obtained, but, in general, these have infinite vertical velocities at the trailing edge of the plate. Consideration of further terms in the expansions suggests a way of canceling the singularity for certain values of the Froude number.

Minimising wave drag for free surface flow past a two-dimensional stern

Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving for the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results found using a forced KdV equation are also presented, as are numerical solutions to the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely.

Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM($s,r$) methods to construct symplectic Runge-Kutta methods for all values of $s$ and $r$ with $s\geq r$. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge-Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

Mapping of multiple quantitative trait loci affecting bovine spongiform encephalopathy

A whole-genome scan was conducted to map quantitative trait loci (QTL) for BSE resistance or susceptibility. Cows from four half-sib families were included and 173 microsatellite markers were used to construct a 2835-cM (Kosambi) linkage map covering 29 autosomes and the pseudoautosomal region of the sex chromosome. Interval mapping by linear regression was applied and extended to a multiple-QTL analysis approach that used identified QTL on other chromosomes as cofactors to increase mapping power. In the multiple-QTL analysis, two genome-wide significant QTL (BTA17 and X/Y ps) and four genome-wide suggestive QTL (BTA1, 6, 13, and 19) were revealed. The QTL identified here using linkage analysis do not overlap with regions previously identified using TDT analysis. One factor that may explain the disparity between the results is that a more extensive data set was used in the present study. Furthermore, methodological differences between TDT and linkage analyses may affect the power of these approaches.

Genome-wide search for markers associated with bovine spongiform encephalopathy

A genome-wide search for markers associated with BSE incidence was performed by using Transmission-Disequilibrium Tests (TDTs). Significant segregation distortion, i.e., unequal transmission probabilities of alleles within a locus, was found for three marker loci on Chromosomes (Chrs) 5, 10, and 20. Although TDTs are robust to false associations owing to hidden population substructures, it cannot distinguish segregation distortion caused by a true association between a marker and bovine spongiform encephalopathy (BSE) from a population-wide distortion. An interaction test and a segregation distortion analysis in half-sib controls were used to disentangle these two alternative hypotheses. None of the markers showed any significant interaction between allele transmission rates and disease status, and only the marker on Chr 10 showed a significant segregation distortion in control individuals. Nevertheless, the control group may have been a mixture of resistant and susceptible but unchallenged individuals. When new genotypes were generated in the vicinity of these three markers, evidence for an association with BSE was confirmed for the locus on Chr 5.

Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods.

Facial emotion modulates the neural mechanisms responsible for short interval time perception

Emotionally arousing events can distort our sense of time. We used mixed block/event-related fMRI design to establish the neural basis for this effect. Nineteen participants were asked to judge whether angry, happy and neutral facial expressions that varied in duration (from 400 to 1,600 ms) were closer in duration to either a short or long duration they learnt previously. Time was overestimated for both angry and happy expressions compared to neutral expressions. For faces presented for 700 ms, facial emotion modulated activity in regions of the timing network Wiener et al. (NeuroImage 49(2):1728–1740, 2010) namely the right supplementary motor area (SMA) and the junction of the right inferior frontal gyrus and anterior insula (IFG/AI). Reaction times were slowest when faces were displayed for 700 ms indicating increased decision making difficulty. Taken together with existing electrophysiological evidence Ng et al. (Neuroscience, doi: 10.3389/fnint.2011.00077, 2011), the effects are consistent with the idea that facial emotion moderates temporal decision making and that the right SMA and right IFG/AI are key neural structures responsible for this effect.