Bit-parallel coprocessor for standard ECC-GF(2m) on FPGA

This paper presents the design of a high-speed coprocessor for Elliptic Curve Cryptography over binary Galois Field (ECC- GF(2m)). The purpose of our coprocessor is to accelerate the scalar multiplication performed over elliptic curve points represented by affine coordinates in polynomial basis. Our method consists of using elliptic curve parameters over GF(2163) in accordance with international security requirements to implement a bit-parallel coprocessor on field-programmable gate-array (FPGA). Our coprocessor performs modular inversion by using a process based on the Stein's algorithm. Results are presented and compared to results of other related works. We conclude that our coprocessor is suitable for comparing with any other ECC-hardware proposal, since its speed is comparable to projective coordinate designs.

Fields of two power conductor

The goal of this work is find a description for fields of two power conductor. By the Kronecker-Weber theorem, these amounts to find the subfields of cyclotomic field $\mathbb{Q}(\xi_{2^r})$, where $\xi_{2^r}$ is a $2^r$-th primitive root of unit and $r$ a positive integer. In this case, the cyclotomic extension isn't cyclic, however its Galois group is generated by two elements and the subfield can be expressed by $\mathbb{Q}(\theta)$ for a $\theta\in\mathbb{Q}(\xi_{2^r})$ convenient.

Hilbert space of curved βγ systems on quadric cones

We clarify the structure of the Hilbert space of curved βγ systems defined by a quadratic constraint. The constraint is studied using intrinsic and BRST methods, and their partition functions are shown to agree. The quantum BRST cohomology is non-empty only at ghost numbers 0 and 1, and there is a one-to-one mapping between these two sectors. In the intrinsic description, the ghost number 1 operators correspond to the ones that are not globally defined on the constrained surface. Extension of the results to the pure spinor superstring is discussed in a separate work.

Sobre certas teorias de cohomologia de grupos e aplicações

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Uma abordagem geométrica para princípios de localização de integrais funcionais

Pós-graduação em Física - IFT

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Algebras determined by their supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes. (C) 2011 Elsevier B.V. All rights reserved.

Intersections on moduli spaces of curves

We present a new approach to perform calculations with the certain standard classes in cohomology of the moduli spaces of curves. It is based on an important lemma of Ionel relating the intersection theoriy of the moduli space of curves and that of the space of admissible coverings. As particular results, we obtain expressions of Hurwitz numbers in terms of the intersections in the tautological ring, expressions of the simplest intersection numbers in terms of Hurwitz numbers, an algorithm of calculation of certain correlators which are the subject of the Witten conjecture, an improved algorithm for intersections related to the Boussinesq hierarchy, expressions for the Hodge integrals over two-pointed ramification cycles, cut-and-join type equations for a large class of intersection numbers, etc.

Gruppi semplici sporadici: i gruppi di Mathieu

La tesi tratta dei gruppi semplici sporadici, in particolar modo dei gruppi di Mathieu. Sono state ripercorse tappe storiche fondamentali, a partire dalla semplicità del gruppo alterno An, n>4, nota a Galois, fino a giungere al teorema di classificazione dei gruppi semplici, di cui i gruppi sporadici rappresentano un caso particolare. Vengono poi proposte diverse costruzioni dei gruppi di Mathieu, passando dall'algebra alla geometria fino alla teoria dell'informazione. Quindi vengono discusse le proprietà principali dei gruppi di Mathieu, e infine si presentano congetture in cui i gruppi di Mathieu, o più in generale i gruppi sporadici, giocano un ruolo fondamentale, come ad esempio nella congettura "moonshine". Al termine della tesi vengono presentati i gruppi di Mathieu in ambiti diversi dal mondo matematico, dal gioco alla musica.

Lagrange-Singularitäten

In dieser Arbeit wird eine Deformationstheorie fürLagrange-Singularitäten entwickelt. Wir definieren einen Komplex von Moduln mit nicht-linearem Differential, densogenannten Lagrange-de Rham-Komplex, dessen ersteKohomologie isomorph zum Raum der infinitesimalenLagrange-Deformationen ist. Wir beschreiben die Beziehung diesesKomplexes zur Theorie der Moduln über dem Ring vonDifferentieloperatoren. Informationen zur Obstruktionstheorie vonLagrange-Deformationen werden aus derzweiten Kohomologie des Lagrange-de Rham-Komplexes gewonnen.Wir zeigen, dass unter einer geometrischen Bedingung an dieSingularität ie Kohomologie von des Lagrange-deRham-Komplexes ausendlich dimensionalen Vektorräumen besteht. Desweiteren wirdeine Methode zur effektiven Berechnung dieser Kohomologie fürquasi-homogene Lagrange-Flächensingularitäten entwickelt. UnterZuhilfenahme von Computeralgebra wird diese Methode für konkreteBeispiele angewendet.

Algebraische Beschreibung von Symmetrien: das Modell der Nichtkommutativen Multi-Tori

In der Nichtkommutativen Geometrie werden Räume und Strukturen durch Algebren beschrieben. Insbesondere werden hierbei klassische Symmetrien durch Hopf-Algebren und Quantengruppen ausgedrückt bzw. verallgemeinert. Wir zeigen in dieser Arbeit, daß der bekannte Quantendoppeltorus, der die Summe aus einem kommutativen und einem nichtkommutativen 2-Torus ist, nur den Spezialfall einer allgemeineren Konstruktion darstellt, die der Summe aus einem kommutativen und mehreren nichtkommutativen n-Tori eine Hopf-Algebren-Struktur zuordnet. Diese Konstruktion führt zur Definition der Nichtkommutativen Multi-Tori. Die Duale dieser Multi-Tori ist eine Kreuzproduktalgebra, die als Quantisierung von Gruppenorbits interpretiert werden kann. Für den Fall von Wurzeln der Eins erhält man wichtige Klassen von endlich-dimensionalen Kac-Algebren, insbesondere die 8-dim. Kac-Paljutkin-Algebra. Ebenfalls für Wurzeln der Eins kann man die Nichtkommutativen Multi-Tori als Hopf-Galois-Erweiterungen des kommutativen Torus interpretieren, wobei die Rolle der typischen Faser von einer endlich-dimensionalen Hopf-Algebra gespielt wird. Der Nichtkommutative 2-Torus besitzt bekanntlich eine u(1)xu(1)-Symmetrie. Wir zeigen, daß er eine größere Quantengruppen-Symmetrie besitzt, die allerdings nicht auf die Spektralen Tripel des Nichtkommutativen Torus fortgesetzt werden kann.

Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

Geometry of BV quantization and Mathai-Quillen formalism

Il formalismo Mathai-Quillen (MQ) è un metodo per costruire la classe di Thom di un fibrato vettoriale attraverso una forma differenziale di profilo Gaussiano. Lo scopo di questa tesi è quello di formulare una nuova rappresentazione della classe di Thom usando aspetti geometrici della quantizzazione Batalin-Vilkovisky (BV). Nella prima parte del lavoro vengono riassunti i formalismi BV e MQ entrambi nel caso finito dimensionale. Infine sfrutteremo la trasformata di Fourier “odd" considerando la forma MQ come una funzione definita su un opportuno spazio graduato.

Risolubilità per radicali

Lo scopo di questa tesi è di esporre il cuore centrale della teoria di Galois, la risolubilità per radicali delle equazioni polinomiali nel caso in cui il campo di partenza abbia caratteristica 0. L’operato è articolato in tre capitoli. Nel primo capitolo vengono introdotte le nozioni fondamentali della teoria dei campi e della teoria di Galois. Nel secondo capitolo, si sviluppa il problema della risolubilità per radicali. Vengono prima introdotti i gruppi risolubili e alcune loro particolarità. Poi vengono introdotte le nozioni di estensioni radicali e risolubili e relativi teoremi. Nel paragrafo 2.3 viene dimostrato il teorema principale della teoria, il teorema di Galois, che identifica la risolubilità del gruppo di Galois con la risolubilità dell’estensione. Infine l’ultimo paragrafo si occupa della risolubilità dei polinomi, sfruttando il loro campo di spezzamento. Nel terzo ed ultimo capitolo, viene discussa la risolubilita` dell’equazione polinomiale generale di grado n. Vengono inoltre riportati diversi esempi ed infine viene presentato un esempio di estensione di Galois di grado primo p non risolubile in caratteristica p.

Frobenius polynomials for Calabi-Yau equations

Sei $\pi:X\rightarrow S$ eine \"uber $\Z$ definierte Familie von Calabi-Yau Varietaten der Dimension drei. Es existiere ein unter dem Gauss-Manin Zusammenhang invarianter Untermodul $M\subset H^3_{DR}(X/S)$ von Rang vier, sodass der Picard-Fuchs Operator $P$ auf $M$ ein sogenannter {\em Calabi-Yau } Operator von Ordnung vier ist. Sei $k$ ein endlicher K\"orper der Charaktetristik $p$, und sei $\pi_0:X_0\rightarrow S_0$ die Reduktion von $\pi$ \uber $k$. F\ur die gew\ohnlichen (ordinary) Fasern $X_{t_0}$ der Familie leiten wir eine explizite Formel zur Berechnung des charakteristischen Polynoms des Frobeniusendomorphismus, des {\em Frobeniuspolynoms}, auf dem korrespondierenden Untermodul $M_{cris}\subset H^3_{cris}(X_{t_0})$ her. Sei nun $f_0(z)$ die Potenzreihenl\osung der Differentialgleichung $Pf=0$ in einer Umgebung der Null. Da eine reziproke Nullstelle des Frobeniuspolynoms in einem Teichm\uller-Punkt $t$ durch $f_0(z)/f_0(z^p)|_{z=t}$ gegeben ist, ist ein entscheidender Schritt in der Berechnung des Frobeniuspolynoms die Konstruktion einer $p-$adischen analytischen Fortsetzung des Quotienten $f_0(z)/f_0(z^p)$ auf den Rand des $p-$adischen Einheitskreises. Kann man die Koeffizienten von $f_0$ mithilfe der konstanten Terme in den Potenzen eines Laurent-Polynoms, dessen Newton-Polyeder den Ursprung als einzigen inneren Gitterpunkt enth\alt, ausdr\ucken,so beweisen wir gewisse Kongruenz-Eigenschaften unter den Koeffizienten von $f_0$. Diese sind entscheidend bei der Konstruktion der analytischen Fortsetzung. Enth\alt die Faser $X_{t_0}$ einen gew\ohnlichen Doppelpunkt, so erwarten wir im Grenz\ubergang, dass das Frobeniuspolynom in zwei Faktoren von Grad eins und einen Faktor von Grad zwei zerf\allt. Der Faktor von Grad zwei ist dabei durch einen Koeffizienten $a_p$ eindeutig bestimmt. Durchl\auft nun $p$ die Menge aller Primzahlen, so erwarten wir aufgrund des Modularit\atssatzes, dass es eine Modulform von Gewicht vier gibt, deren Koeffizienten durch die Koeffizienten $a_p$ gegeben sind. Diese Erwartung hat sich durch unsere umfangreichen Rechnungen best\atigt. Dar\uberhinaus leiten wir weitere Formeln zur Bestimmung des Frobeniuspolynoms her, in welchen auch die nicht-holomorphen L\osungen der Gleichung $Pf=0$ in einer Umgebung der Null eine Rolle spielen.