Efficient SIMD arithmetic modulo a Mersenne number

This paper describes carry-less arithmetic operations modulo an integer 2^M − 1 in the thousand-bit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization

Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page

An Analysis of the Common Core State Standards for Mathematics and the Content of Math 095: Essentials of Algebra at the University of Wisconsin-Milwaukee

In this analysis we present the content in Math 095: Essentials of Algebra at the University of Wisconsin-Milwaukee that is aligned to the Common Core State Standards for Mathematics. We find that the content in Math 095 is aligned to a small subset of the high school Number and Quantity, Algebra, and Function standards. We present a representative sample of homework and assessment items from the traditional lecture format of Math 095 and compare them to assessment items released by the Smarter Balanced Assessment Consortium and Illustrative Mathematics. We then discuss content from the Common Core State Standards for Mathematics that is absent from Math 095 or overlaps with Math 105: Intermediate Algebra. We conclude that overall the content from Math 095 meets the standards that require students to solve, graph, factor, rewrite expressions, and apply operations to expressions; however, we find that standards expecting students to understand concepts, model with mathematics, and explain reasoning are not met in the course content. We conclude by presenting recommendations based on our findings

Mersenne Factorization Factory

We present work in progress to completely factor seventeen Mersenne numbers using a variant of the special number field sieve where sieving on the algebraic side is shared among the numbers. It is expected that it reduces the overall factoring effort by more than 50%. As far as we know this is the first practical application of Coppersmith’s “factorization factory” idea. Most factorizations used a new double-product approach that led to additional savings in the matrix step

On the factorization of polynomials over algebraic fields

SIGLEAvailable from British Library Document Supply Centre- DSC:DX86869 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Counting walks with large steps in an orthant

International audienceIn the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions ofquadrant walks are now well-understood, provided the allowed steps aresmall, that is $S \subset \{-1, 0,1\}^2$. In particular, having smallsteps is crucial for the definition of a certain group of bi-rationaltransformations of the plane. It has been proved that this group is finite ifand only if the corresponding generating function is D-finite (that is, it satisfies a lineardifferential equation with polynomial coefficients). This group is also thekey to the uniform solution of 19 of the 23 small step models possessing afinite group.In contrast, almost nothing is known for walks with arbitrary steps. In thispaper, we extend the definition of the group, or rather of the associatedorbit, to this general case, and generalize the above uniform solution ofsmall step models. When this approach works, it invariably yields a D-finitegenerating function. We apply it to many quadrant problems, including some infinite families.After developing the general theory, we consider the $13\ 110$ two-dimensionalmodels with steps in $\{-2,-1,0,1\}^2$ having at least one $-2$ coordinate. Weprove that only 240 of them have a finite orbit, and solve 231 of them withour method. The 9 remaining models are the counterparts of the 4 models of thesmall step case that resist the uniform solution method (and which are knownto have an algebraic generating function). We conjecture D-finiteness for their generatingfunctions, but only two of them are likely to be algebraic. We also provenon-D-finiteness for the $12\ 870$ models with an infinite orbit, except for16 of them

Harnessing the power of GPUs for problems in real algebraic geometry

This thesis presents novel parallel algorithms to leverage the power of GPUs (Graphics Processing Units) for exact computations with polynomials having large integer coefficients. The significance of such computations, especially in real algebraic geometry, is hard to undermine. On massively-parallel architectures such as GPU, the degree of datalevel parallelism exposed by an algorithm is the main performance factor. We attain high efficiency through the use of structured matrix theory to assist the realization of relevant operations on polynomials on the graphics hardware. A detailed complexity analysis, assuming the PRAM model, also confirms that our approach achieves a substantially better parallel complexity in comparison to classical algorithms used for symbolic computations. Aside from the theoretical considerations, a large portion of this work is dedicated to the actual algorithm development and optimization techniques where we pay close attention to the specifics of the graphics hardware. As a byproduct of this work, we have developed high-throughput modular arithmetic which we expect to be useful for other GPU applications, in particular, open-key cryptography. We further discuss the algorithms for the solution of a system of polynomial equations, topology computation of algebraic curves and curve visualization which can profit to the full extent from the GPU acceleration. Extensive benchmarking on a real data demonstrates the superiority of our algorithms over several state-of-the-art approaches available to date. This thesis is written in English.Diese Arbeit beschäftigt sich mit neuen parallelen Algorithmen, die das Leistungspotenzial der Grafik-Prozessoren (GPUs) zur exakten Berechnungen mit ganzzahlige Polynomen nutzen. Solche symbolische Berechnungen sind von großer Bedeutung zur Lösung vieler Probleme aus der reellen algebraischen Geometrie. Für die effziente Implementierung eines Algorithmus auf massiv-parallelen Hardwarearchitekturen, wie z.B. GPU, ist vor allem auf eine hohe Datenparallelität zu achten. Unter Verwendung von Ergebnissen aus der strukturierten Matrix-Theorie konnten wir die entsprechenden Operationen mit Polynomen auf der Grafikkarte leicht übertragen. Außerdem zeigt eine Komplexitätanalyse im PRAM-Rechenmodell, dass die von uns entwickelten Verfahren eine deutlich bessere Komplexität aufweisen als dies für die klassischen Verfahren der Fall ist. Neben dem theoretischen Ergebnis liegt ein weiterer Schwerpunkt dieser Arbeit in der praktischen Implementierung der betrachteten Algorithmen, wobei wir auf der Besonderheiten der Grafikhardware achten. Im Rahmen dieser Arbeit haben wir hocheffiziente modulare Arithmetik entwickelt, von der wir erwarten, dass sie sich für andere GPU Anwendungen, insbesondere der Public-Key-Kryptographie, als nützlich erweisen wird. Darüber hinaus betrachten wir Algorithmen für die Lösung eines Systems von Polynomgleichungen, Topologie Berechnung der algebraischen Kurven und deren Visualisierung welche in vollem Umfang von der GPU-Leistung profitieren können. Zahlreiche Experimente belegen dass wir zur Zeit die beste Verfahren zur Verfügung stellen. Diese Dissertation ist in englischer Sprache verfasst

The Riemann zeta function and zeta regularization in Casimir effect

[EN] This work is divided into three chapters. The aim of Chapter 1 is to introduce some basic tools that will be used throughout the rest of the dissertation, in order to make it as self-contained as possible. In Chapter 2, an elementary overview of the main features of the Riemann zeta function is presented. In Chapter 3, we introduce the more general concept of zeta function associated with a differential operator, showing that it can be used as a summation method for divergent series. Finally, we introduce the Casimir effect and we compute the value of the Casimir force in the simplest scenario.[ES] Este trabajo se divide en tres capítulos. El objetivo del Capítulo 1 es introducir algunas herramientas básicas que se utilizarán durante el resto de la disertación, para que sea lo más autónoma posible. En el capítulo 2, una descripción general elemental de las características principales de la función zeta de Riemann es presentado. En el Capítulo 3, presentamos el concepto más general de función zeta asociada con un operador diferencial, mostrando que se puede usar como un método de suma para series divergentes. Finalmente, presentamos el Casimir efecto y calculamos el valor de la fuerza de Casimir en el escenario más simple