Thèses : « Finite fields (Algebra) Geometry »

1

Rovi, Carmen. « Algebraic Curves over Finite Fields ». Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.

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This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of N_q(g) is now known.

At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

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Berardini, Elena. « Algebraic geometry codes from surfaces over finite fields ». Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0170.

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Nous proposons, dans cette thèse, une étude théorique des codes géométriques algébriques construits à partir de surfaces définies sur les corps finis. Nous prouvons des bornes inférieures pour la distance minimale des codes sur des surfaces dont le diviseur canonique est soit nef soit anti-strictement nef et sur des surfaces sans courbes irréductibles de petit genre. Nous améliorons ces bornes inférieures dans le cas des surfaces dont le nombre de Picard arithmétique est égal à un, des surfaces sans courbes de petite auto-intersection et des surfaces fibrées. Ensuite, nous appliquons ces bornes aux surfaces plongées dans P3. Une attention particulière est accordée aux codes construits à partir des surfaces abéliennes. Dans ce contexte, nous donnons une borne générale sur la distance minimale et nous démontrons que cette estimation peut être améliorée en supposant que la surface abélienne ne contient pas de courbes absolument irréductibles de petit genre. Dans cette optique nous caractérisons toutes les surfaces abéliennes qui ne contiennent pas de courbes absolument irréductibles de genre inférieur ou égal à 2. Cette approche nous conduit naturellement à considérer les restrictions de Weil de courbes elliptiques et les surfaces abéliennes qui n'admettent pas de polarisation principale
In this thesis we provide a theoretical study of algebraic geometry codes from surfaces defined over finite fields. We prove lower bounds for the minimum distance of codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Then we apply these bounds to surfaces embedded in P3. A special attention is given to codes constructed from abelian surfaces. In this context we give a general bound on the minimum distance and we prove that this estimation can be sharpened under the assumption that the abelian surface does not contain absolutely irreducible curves of small genus. In this perspective we characterize all abelian surfaces which do not contain absolutely irreducible curves of genus up to 2. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization

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Jogia, Danesh Michael Mathematics &amp Statistics Faculty of Science UNSW. « Algebraic aspects of integrability and reversibility in maps ». Publisher:University of New South Wales. Mathematics & ; Statistics, 2008. http://handle.unsw.edu.au/1959.4/40947.

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We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.

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Grout, Jason Nicholas. « The Minimum Rank Problem Over Finite Fields ». Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1995.pdf.

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Hart, Derrick. « Explorations of geometric combinatorics in vector spaces over finite fields ». Diss., Columbia, Mo. : University of Missouri-Columbia, 2008. http://hdl.handle.net/10355/5585.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2008.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 8, 2009) Vita. Includes bibliographical references.

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Culbert, Craig W. « Spreads of three-dimensional and five-dimensional finite projective geometries ». Access to citation, abstract and download form provided by ProQuest Information and Learning Company ; downloadable PDF file, 101 p, 2009. http://proquest.umi.com/pqdweb?did=1891555371&sid=3&Fmt=2&clientId=8331&RQT=309&VName=PQD.

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Marseglia, Stefano. « Isomorphism classes of abelian varieties over finite fields ». Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-130316.

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Deligne and Howe described polarized abelian varieties over finite fields in terms of finitely generated free Z-modules satisfying a list of easy to state axioms. In this thesis we address the problem of developing an effective algorithm to compute isomorphism classes of (principally) polarized abelian varieties over a finite field, together with their automorphism groups. Performing such computations requires the knowledge of the ideal classes (both invertible and non-invertible) of certain orders in number fields. Hence we describe a method to compute the ideal class monoid of an order and we produce concrete computations in dimension 2, 3 and 4.

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Castilho, Tiago Nunes 1983. « Sobre o numero de pontos racionais de curvas sobre corpos finitos ». [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307074.

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Orientador: Fernando Eduardo Torres Orihuela
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-10T15:12:25Z (GMT). No. of bitstreams: 1 Castilho_TiagoNunes_M.pdf: 813127 bytes, checksum: 313e9951b003dcd0e0876813659d7050 (MD5) Previous issue date: 2008
Resumo: Nesta dissertacao estudamos cotas para o numero de pontos racionais de curvas definidas sobre corpos finitos tendo como ponto de partida a teoria de Stohr-Voloch
Abstract: In this work we study upper bounds on the number of rational points of curves over finite fields by using the Stohr-Voloch theory
Mestrado
Algebra Comutativa, Geometria Algebrica
Mestre em Matemática

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Ribeiro, Beatriz Casulari da Motta 1984. « O arco associado a uma generalização da curva Hermitiana ». [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307081.

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Orientadores: Fernando Eduardo Torres Orihuela, Herivelto Martins Borges Filho
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-19T05:54:38Z (GMT). No. of bitstreams: 1 Ribeiro_BeatrizCasularidaMotta_D.pdf: 51476410 bytes, checksum: 46cb0c7a6206a5f0683b23a73ff3938e (MD5) Previous issue date: 2011
Resumo: Obtemos novos arcos completos associados ao conjunto de pontos racionais de uma certa generalização da curva Hermitiana que é Frobenius não-clássica. A construção está relacionada ao cálculo do número de pontos racionais de uma classe de curvas de Artin-Schreier
Abstract: We obtain new complete arcs arising from the set of rational points of a certain generalization of the Hermitian plane curve which is Frobenius non-classical. Our construction is related to the computation of the number of rational points of a class of Artin-Schreier curves
Doutorado
Matematica
Doutor em Matemática

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Albuquerque, JoÃo Victor Maximiano. « Finitude do grupo das classes de um corpo de nÃmeros via empacotamentos reticulados ». Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11247.

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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
Este trabalho Ã baseado no artigo Finiteness of the class group of a number field via lattice packings. Daremos aqui uma prova alternativa da finitude do grupo das classes de um corpo de nÃmeros de grau n. Ela Ã baseada apenas no fato de que a densidade de centro de um empacotamento reticulado n-dimensional Ã limitado fora do infinito.
This work is based on the article Finiteness of the class group of a number field via lattice packings. An alternative proof of the finiteness of the class group of a number field of the degree n is presented. It is based solely on the fact that the center density of an n-dimensional lattice packing is bounded away from infinity.

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Bergvall, Olof. « Cohomology of arrangements and moduli spaces ». Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-132822.

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This thesis mainly concerns the cohomology of the moduli spaces ℳ3[2] and ℳ3,1[2] of genus 3 curves with level 2 structure without respectively with a marked point and some of their natural subspaces. A genus 3 curve which is not hyperelliptic can be realized as a plane quartic and the moduli spaces 𝒬[2] and 𝒬1[2] of plane quartics without respectively with a marked point are given special attention. The spaces considered come with a natural action of the symplectic group Sp(6,𝔽2) and their cohomology groups thus become Sp(6,𝔽2)-representations. All computations are therefore Sp(6,𝔽2)-equivariant. We also study the mixed Hodge structures of these cohomology groups. The computations for ℳ3[2] are mainly via point counts over finite fields while the computations for ℳ3,1[2] primarily uses a description due to Looijenga in terms of arrangements associated to root systems. This leads us to the computation of the cohomology of complements of toric arrangements associated to root systems. These varieties come with an action of the corresponding Weyl group and the computations are equivariant with respect to this action.

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Amorós, Carafí Laia. « Images of Galois representations and p-adic models of Shimura curves ». Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/471452.

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The Langlands program is a vast and unifying network of conjectures that connect the world of automorphic representations of reductive algebraic groups and the world of Galois representations. These conjectures associate an automorphic representation of a reductive algebraic group to every n-dimensional representation of a Galois group, and the other way around: they attach a Galois representation to any automorphic representation of a reductive algebraic group. Moreover, these correspondences are done in such a way that the automorphic L-functions attached to the two objects coincide. The theory of modular forms is a field of complex analysis whose main importance lies on its connections and applications to number theory. We will make use, on the one hand, of the arithmetic properties of modular forms to study certain Galois representations and their number theoretic meaning. On the other hand, we will use the geometric meaning of these complex analytic functions to study a natural generalization of modular curves. A modular curve is a geometric object that parametrizes isomorphism classes of elliptic curves together with some additional structure depending on some modular subgroup. The generalization that we will be interested in are the so called Shimura curves. We will be particularly interested in their p-adic models. In this thesis, we treat two different topics, one in each side of the Langlands program. In the Galois representations' side, we are interested in Galois representations that take values in local Hecke algebras attached to modular forms over finite fields. In the automorphic forms' side, we are interested in Shimura curves: we develop some arithmetic results in definite quaternion algebras and give some results about Mumford curves covering p-adic Shimura curves.

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Marangoni, Davide. « On Derived de Rham cohomology ». Thesis, Bordeaux, 2020. http://www.theses.fr/2020BORD0095.

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La cohomologie de de Rham dérivée a été introduite par Luc Illusie en 1972, suite à ses travaux sur le complexe cotangent. Cette théorie semble avoir été oubliée jusqu’aux travaux récents de Bhatt et Beilinson, qui ont donné diverses applications, notamment en théorie de Hodge p-adique. D’autre part, la cohomologie de Rham dérivée intervient de manière cruciale dans une conjecture de Flach-Morin sur les valeurs spéciales des fonctions zêta des schémas arithmétiques. Dans cette thèse, on se propose d’étudier et de calculer la cohomologie de de Rham dérivée dans certains cas
The derived de Rham complex has been introduced by Illusie in 1972. Its definition relies on the notion of cotangent complex. This theory seems to have been forgot until the recents works by Be˘ılinson and Bhatt, who gave several applications, in particular in p-adic Hodge Theory. On the other hand, the derived de Rham cohomology has a crucial role in a conjecture by Flach-Morin about special values of zeta functions for arithmetic schemes. The aim of this thesis is to study and compute the Hodge completed derived de Rham complex in some cases

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Wilcox, Nicholas. « A Computational Introduction to Elliptic and Hyperelliptic Curve Cryptography ». Oberlin College Honors Theses / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1528649455201473.

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Gordon, Neil Andrew. « Finite geometry and computer algebra, with applications ». Thesis, University of Hull, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.262412.

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Giuzzi, Luca. « Hermitian varieties over finite fields ». Thesis, University of Sussex, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.326913.

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Prešern, Mateja. « Existence problems of primitive polynomials over finite fields ». Connect to e-thesis. Move to record for print version, 2007. http://theses.gla.ac.uk/50/.

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Thesis (Ph.D.) - University of Glasgow, 2007.
Ph.D. thesis submitted to the Department of Mathematics, Faculty of Information and Mathematical Sciences, University of Glasgow, 2007. Includes bibliographical references.

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GOMEZ-CALDERON, JAVIER. « POLYNOMIALS WITH SMALL VALUE SET OVER FINITE FIELDS ». Diss., The University of Arizona, 1986. http://hdl.handle.net/10150/183933.

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Let K(q) be the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coefficients in K(q). Let C(f) denote the number of distinct values of f(x) as x ranges over K(q). It is easy to show that C(f) ≤ [|(q - 1)/d|] + 1. Now, there is a characterization of polynomials of degree d < √q for which C(f) = [|(q - 1)/d|] +1. The main object of this work is to give a characterization for polynomials of degree d < ⁴√q for which C(f) < 2q/d. Using two well known theorems: Hurwitz genus formula and Andre Weil's theorem, the Riemann Hypothesis for Algebraic Function Fields, it is shown that if d < ⁴√q and C(f) < 2q/d then f(x) - f(y) factors into at least d/2 absolutely irreducible factors and f(x) has one of the following forms: (UNFORMATTED TABLE FOLLOWS) f(x - λ) = D(d,a)(x) + c, d|(q² - 1), f(x - λ) = D(r,a)(∙ ∙ ∙ ((x²+b₁)²+b₂)²+ ∙ ∙ ∙ +b(m)), d|(q² - 1), d=2ᵐ∙r, and (2,r) = 1 f(x - λ) = (x² + a)ᵈ/² + b, d/2|(q - 1), f(x - λ) = (∙ ∙ ∙((x²+b₁)²+b₂)² + ∙ ∙ ∙ +b(m))ʳ+c, d|(q - 1), d=2ᵐ∙r, f(x - λ) = xᵈ + a, d|(q - 1), f(x - λ) = x(x³ + ax + b) + c, f(x - λ) = x(x³ + ax + b) (x² + a) + e, f(x - λ) = D₃,ₐ(x² + c), c² ≠ 4a, f(x - λ) = (x³ + a)ⁱ + b, i = 1, 2, 3, or 4, f(x - λ) = x³(x³ + a)³ +b, f(x - λ) = x⁴(x⁴ + a)² +b or f(x - λ) = (x⁴ + a) ⁱ + b, i = 1,2 or 3, where D(d,a)(x) denotes the Dickson’s polynomial of degree d. Finally to show other polynomials with small value set, the following equation is obtained C((fᵐ + b)ⁿ) = αq/d + O(√q) where α = (1 – (1 – 1/m)ⁿ)m and the constant implied in O(√q) is independent of q.

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ASSIS, FRANCISCO MARCOS DE. « DECODING OF ALGEBRAIC GEOMETRY CODES AND THE USE OF NEURAL NETWORKS FOR FINITE FIELD ». PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1994. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8517@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Este trabalho propõe um algoritmo para decodificação de códigos de geometria algébrica. Usando as propriedades geométricas da curva que define um código de Goppa com distância projetada d, método permite decodificar até [d - 1/ 2] erros em palavra recebida, sem esforço computacional adicional. As curvas de F. K. Schimdt são usada para construir uma nova classe de códigos de geometria algébrica, algumas propriedades destes novos códigos são apresentadas. Redes neurais não ortodoxas do tipo feedforward e não treinadas são usadas para construir circuitos que permitem calcular logaritmos de Zech eficientemente e, portanto, realizar aritmética em corpos finitos sem uso de tabelas.
A method for decoding algbraic geometric codes is proposed. By using geometric properties of the curve defining a Goppa code, with projected distance d the algorithm corrects until [d - 1 / 2 ] errors without additional computational cost. F. K. Schmidt curves are used in construction of a new class of algebric geometric error correcting codes. A feedfoward neural network is proposed that realizes a efficient Zech`s logarithms calculation. The neural network proposed is non-ortodoxal in sense that non- training is used for these construction.

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Akleylek, Sedat. « On The Representation Of Finite Fields ». Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612727/index.pdf.

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The representation of field elements has a great impact on the performance of the finite field arithmetic. In this thesis, we give modified version of redundant representation which works for any finite fields of arbitrary characteristics to design arithmetic circuits with small complexity. Using our modified redundant representation, we improve many of the complexity values. We then propose new representations as an alternative way to represent finite fields of characteristic two by using Charlier and Hermite polynomials. We show that multiplication in these representations can be achieved with subquadratic space complexity. Charlier and Hermite representations enable us to find binomial, trinomial or quadranomial irreducible polynomials which allows us faster modular reduction over binary fields when there is no desirable such low weight irreducible polynomial in other representations. These representations are very interesting for the NIST and SEC recommended binary fields GF(2^{283}) and GF(2^{571}) since there is no optimal normal basis (ONB) for the corresponding extensions. It is also shown that in some cases the proposed representations have better space complexity even if there exists an ONB for the corresponding extension.

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Frisk, Dubsky Brendan. « Classication of simple complex weight modules with finite-dimensional weight spaces over the Schrödinger algebra ». Thesis, Uppsala universitet, Algebra och geometri, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-200606.

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Majid, Shahn, et Andreas Cap@esi ac at. « Riemannian Geometry of Quantum Groups and Finite Groups with ». ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi902.ps.

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Karaoglu, Fatma. « The cubic surfaces with twenty-seven lines over finite fields ». Thesis, University of Sussex, 2018. http://sro.sussex.ac.uk/id/eprint/78533/.

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In this thesis, we classify the cubic surfaces with twenty-seven lines in three dimensional projective space over small finite fields. We use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3; q) from 6-arcs not on a conic in PG(2; q). We introduce computational and geometrical procedures for the classification of cubic surfaces over the finite field Fq. The performance of the algorithms is illustrated by the example of cubic surfaces over F13, F17 and F19.

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Sistko, Alexander Harris. « Maximal subalgebras of finite-dimensional algebras : with connections to representation theory and geometry ». Diss., University of Iowa, 2019. https://ir.uiowa.edu/etd/6857.

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Let $k$ be a field and $B$ a finite-dimensional, associative, unital $k$-algebra. For each $1 \le d \le \dim_kB$, let $\operatorname{AlgGr}_d(B)$ denote the projective variety of $d$-dimensional subalgebras of $B$, and let $\operatorname{Aut}_k(B)$ denote the automorphism group of $B$. In this thesis, we are primarily concerned with understanding the relationship between $\operatorname{AlgGr}_d(B)$, the representation theory of $B$, and the representation theory of $\operatorname{Aut}_k(B)$. We begin by proving fundamental structure theorems for the maximal subalgebras of $B$. We show that maximal subalgebras of $B$ come in two flavors, which we call split type and separable type. As a consequence, we provide complete classifications for maximal subalgebras of semisimple algebras and basic algebras. We also demonstrate that the maximality of $A$ in $B$ is related to the representation theory of $B$, through the separability of functors closely associated with the extension $A \subset B$. The rest of this document showcases applications of these results. For $k = \bar{k}$, we compute the maximal dimension of a proper subalgebra of $B$. We discuss the problem of computing the minimal number of generators for $B$ (as an algebra), and provide upper and lower bounds for basic algebras. We then study $\operatorname{AlgGr}_d(B)$ in detail, again when $B$ is basic. When $d = \dim_kB-1$, we find a projective embedding of $\operatorname{AlgGr}_d(B)$, and explicitly describe its associated homogeneous vanishing ideal. In turn, we provide a simple description of its irreducible components. We find equivalent conditions for this variety to be a finite union of $\operatorname{Aut}_k(B)$-orbits, and describe several classes of algebras which satisfy these conditions. Furthermore, we provide an algebraic description for the orbits of connected maximal subalgebras of type-$\mathbb{A}$ path algebras. Finally, we study the fixed-point variety $\operatorname{AlgGr}_d(B)^{\operatorname{Aut}_k(B)}$ (for general $d$), which connects naturally to the representation theory of $\operatorname{Aut}_k(B)$. We investigate the case where $B$ is a truncated path algebra over $\mathbb{C}$ in detail.

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Veliz-Cuba, Alan A. « The Algebra of Systems Biology ». Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/28240.

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In order to understand biochemical networks we need to know not only how their parts work but also how they interact with each other. The goal of systems biology is to look at biological systems as a whole to understand how interactions of the parts can give rise to complex dynamics. In order to do this efficiently, new techniques have to be developed. This work shows how tools from mathematics are suitable to study problems in systems biology such as modeling, dynamics prediction, reverse engineering and many others. The advantage of using mathematical tools is that there is a large number of theory, algorithms and software available. This work focuses on how algebra can contribute to answer questions arising from systems biology.
Ph. D.

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Vargas, Jorge Ivan. « A characterization of pseudo-orders in the ring Zn ». To access this resource online via ProQuest Dissertations and Theses @ UTEP, 2009. http://0-proquest.umi.com.lib.utep.edu/login?COPT=REJTPTU0YmImSU5UPTAmVkVSPTI=&clientId=2515.

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Caullery, Florian. « Polynomes sur les corps finis pour la cryptographie ». Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4013/document.

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Les fonctions de F_q dans lui-même sont des objets étudiés dans de divers domaines tels que la cryptographie, la théorie des codes correcteurs d'erreurs, la géométrie finie ainsi que la géométrie algébrique. Il est bien connu que ces fonctions sont en correspondance exacte avec les polynômes en une variable à coefficients dans F_q. Nous étudierons trois classes de polynômes particulières: les polynômes Presque Parfaitement Non linéaires (Almost Perfect Nonlinear (APN)), les polynômes planaires ou parfaitement non linéaire (PN) et les o-polynômes.Les fonctions APN sont principalement étudiées pour leurs applications en cryptographie. En effet, ces fonctions sont celles qui offre la meilleure résistance contre la cryptanalyse différentielle.Les polynômes PN et les o-polynômes sont eux liés à des problèmes célèbres de géométrie finie. Les premiers décrivent des plans projectifs et les seconds sont en correspondance directe avec les ovales et hyperovales de P^2(F_q). Néanmoins, leurs champ d'application a été récemment étendu à la cryptographie symétrique et à la théorie des codes correcteurs d'erreurs.L'un des moyens utilisé pour compléter la classification est de considérer les polynômes présentant l'une des propriétés recherchées sur une infinité d'extension de F_q. Ces fonctions sont appelées fonction APN (respectivement PN ou o-polynômes) exceptionnelles.Nous étendrons la classification des polynômes APN et PN exceptionnels et nous donneront une description complète des o-polynômes exceptionnels. Les techniques employées sont basées principalement sur la borne de Lang-Weil et sur des méthodes élémentaires
Functions from F_q to itself are interesting objects arising in various domains such as cryptography, coding theory, finite geometry or algebraic geometry. It is well known that these functions admit a univariate polynomial representation. There exists many interesting classes of such polynomials with plenty of applications in pure or applied maths. We are interested in three of them: Almost Perfect Nonlinear (APN) polynomials, Planar (PN) polynomials and o-polynomials. APN polynomials are mostly used in cryptography to provide S-boxes with the best resistance to differential cryptanalysis and in coding theory to construct double error-correcting codes. PN polynomials and o-polynomials first appeared in finite geometry. They give rise respectively to projective planes and ovals in P^2(F_q). Also, their field of applications was recently extended to symmetric cryptography and error-correcting codes.A complete classification of APN, PN and o-polynomials is an interesting open problem that has been widely studied by many authors. A first approach toward the classification was to consider only power functions and the studies were recently extended to polynomial functions.One way to face the problem of the classification is to consider the polynomials that are APN, PN or o-polynomials over infinitely many extensions of F_q, namely, the exceptional APN, PN or o-polynomials.We improve the partial classification of exceptional APN and PN polynomials and give a full classification of exceptional o-polynomials. The proof technique is based on the Lang-Weil bound for the number of rational points in algebraic varieties together with elementary methods

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Psioda, Matthew. « An examination of the structure of extension families of irreducible polynomials over finite fields / ». Electronic version (PDF), 2006. http://dl.uncw.edu/etd/2006/psiodam/matthewpsioda.pdf.

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Baktir, Selcuk. « Efficient algorithms for finite fields, with applications in elliptic curve cryptography ». Link to electronic thesis, 2003. http://www.wpi.edu/Pubs/ETD/Available/etd-0501103-132249.

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Thesis (M.S.)--Worcester Polytechnic Institute.
Keywords: multiplication; OTF; optimal extension fields; finite fields; optimal tower fields; cryptography; OEF; inversion; finite field arithmetic; elliptic curve cryptography. Includes bibliographical references (p. 50-52).

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Cam, Vural. « Drinfeld Modular Curves With Many Rational Points Over Finite Fields ». Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613118/index.pdf.

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In our study Fq denotes the finite field with q elements. It is interesting to construct curves of given genus over Fq with many Fq -rational points. Drinfeld modular curves can be used to construct that kind of curves over Fq . In this study we will use reductions of the Drinfeld modular curves X_{0} (n) to obtain curves over finite fields with many rational points. The main idea is to divide the Drinfeld modular curves by an Atkin-Lehner involution which has many fixed points to obtain a quotient with a better #{rational points} /genus ratio. If we divide the Drinfeld modular curve X_{0} (n) by an involution W, then the number of rational points of the quotient curve WX_{0} (n) is not less than half of the original number. On the other hand, if this involution has many fixed points, then by the Hurwitz-Genus formula the genus of the curve WX_{0} (n) is much less than half of the g (X_{0}(n)).

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Bosa, Puigredon Joan. « Continuous fields of c-algebras, their cuntz semigroup and the geometry of dimension fuctions ». Doctoral thesis, Universitat Autònoma de Barcelona, 2013. http://hdl.handle.net/10803/126516.

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Aquesta tesi doctoral tracta sobre C*-àlgebres i els seus invariants de Teoria K. Ens hem centrat principalment en l’estructura d’una classe de C*-àlgebres anomenada camps continus i l’estudi d’un dels seus invariants: el semigrup de Cuntz. Més concretament, analitzem el següent: (1)- Estructura dels camps continus : A la literatura hi ha dos exemples que donen una idea clara sobre la complexitat dels camps continus de C*-àlgebres. El primer va ser construït per M. Dadarlat i G. A. Elliott al 2007 i és un camp continu A sobre l’interval unitat amb fibres mútuament isomorfes, Teoria K no finitament generada i que no és localment trivial enlloc. El segon exemple mostra que, fins i tot quan la Teoria K de les fibres s’anul·la, el camp pot ser no trivial enlloc si l’espai base té dimensió infinita (Dadarlat, 2009). Veient aquests exemples és natural preguntar-se quina és l’estructura dels camps continus d’àlgebres de Kirchberg sobre un espai de dimensió finita, amb fibres mútuament isomorfes i Teoria K finitament generada. Tractem aquesta qüestió al Capítol 2 de la memòria. (2)- El semigrup de Cuntz de camps continus : Per a C*-àlgebres de dimensió baixa sense obstruccions cohomològiques, una descripció del seu semigrup de Cuntz, a través d’avaluació puntual, s’ha obtingut en termes de funcions semicontínues sobre l’expectre que prenen valors en els enters positius estesos (Robert, 2009). Per camps més generals la clau està en descriure l’aplicació següent: _: Cu(A) ! Q x2X Cu(Ax) donada per _hai = (ha(x)i)x2X; on Cu(Ax) és el semigrup de Cuntz de la fibra Ax. En el Capítol 3 de la memòria, l’aplicació _ s’estudia en el cas que X tingui dimensió petita i totes les fibres de la C(X)-àlgebra A no són necessàriament isomorfes entre sí. Més concretament, demostrem que és possible recuperar el semigrup de Cuntz d’una classe adequada de camps continus com el semigrup de seccions globals de tx2XCu(Ax) a X. Això s’utilitza posteriorment per reescriure un resultat de classificació degut a Dadarlat, Elliott i Niu (2012) utilitzant un sol invariant en comptes d’un feix de grups. (3)-Funcions de dimensió en una C*-algebra : L’estudi de funcions de dimensió va ser iniciat per Cuntz a 1978, i desenvolupat posteriorment per Blackadar i Handelman al 1982. En el seu article van aparèixer dues preguntes naturals: decidir si l’espai afí de funcions de dimensió és un símplex, i si també el conjunt de funcions de dimensió semicontínues inferiorment és dens a l’espai de totes les funcions de dimensió. En el Capítol 4 calculem el rang estable d’algunes classes de camps continus i això ens ajuda a provar que les dues conjectures anteriors tenen resposta afirmativa per camps continus A sobre espais de dimensió 1 i amb hipòtesis febles en les seves fibres.
This thesis deals with C*-algebras and their K-theoretical invariants. We have mainly focused on the structure of a class of C*-algebras called continuous fields, and the study of one of its invariants, the Cuntz semigroup. More concretely, we analyse the following: (1)-Structure of Continuous Fields of C*-algebras : In the literature there are two examples which clearly give an idea about the complexity of continuous field C*-algebras. The first one was constructed by M. Dadarlat and G. A. Elliott in 2007, and it is a continuous field C*- algebra A over the unit interval with mutually isomorphic fibers, with non-finitely generated K-theory and such that it is nowhere locally trivial. The second example shows that, even if the K-theory of the fibers vanish, the field can be nowhere locally trivial if the base space is infinite-dimensional (Dadarlat, 2009). From the above examples, it is natural to ask which is the structure of continuous fields of Kirchberg algebras over a finite-dimensional space with mutually isomorphic fibers and finitely generated K-theory. This question has been adressed in Chapter 2 of the memoir. (2)-The Cuntz semigroup of continuous field C*-algebras : For commutative C*-algebras of lower dimension where there are no cohomological obstructions, a description of their Cuntz semigroup via point evaluation has been obtained in terms of (extended) integer valued lower semicontinuous functions on their spectrum (Robert, 2009). For more general continuous fields, the key is to describe the map : Cu(A) ! Q x2X Cu(Ax) given by hai = (ha(x)i)x2X; where Cu(Ax) is the Cuntz semigroup of the fiber Ax. In Chapter 3 of the memoir, the map is studied in the case when X has low dimension and all the fibers of the C(X)-algebra A are not necessarily mutually isomorphic. Concretely, we prove that it is possible to recover the Cuntz semigroup of a suitable class of continuous fields as the semigroup of global sections of tx2XCu(Ax) to X. This is further used to rephrase a classification result by Dadarlat, Elliott and Niu (2012) by using a single invariant instead of a sheaf of groups. (3)-Dimension Functions on a C*-algebra : The study of dimension functions on C -algebras was started by Cuntz in 1978, and further developed by B. Blackadar and D. Handelman in 1982. In the latter article, two natural questions arised: to decide whether the affine space of dimension functions is a simplex, and also whether the set of lower semicontinuous dimension functions is dense in the space of all dimension functions. In Chapter 4 we compute the stable rank of some class of continuous field C*-algebras, which helps us to move on to show that the above two conjectures have affirmative answers for continuous fields A over one-dimensional spaces and with mild assumptions on their fibers.

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Park, Hong Goo. « Polynomial Isomorphisms of Cayley Objects Over a Finite Field ». Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc331144/.

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In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.

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Kurtaran, Ozbudak Elif. « Results On Some Authentication Codes ». Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/2/12610350/index.pdf.

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In this thesis we study a class of authentication codes with secrecy. We obtain the maximum success probability of the impersonation attack and the maximum success probability of the substitution attack on these authentication codes with secrecy. Moreover we determine the level of secrecy provided by these authentication codes. Our methods are based on the theory of algebraic function fields over finite fields. We study a certain class of algebraic function fields over finite fields related to this class of authentication codes. We also determine the number of rational places of this class of algebraic function fields.

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Solanki, Nikesh. « Uniform companions for expansions of large differential fields ». Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/uniform-companions-for-expansions-of-large-differential-fields(a565a0d0-24b5-40a6-a414-5ead1631bc8d).html.

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Ranorovelonalohotsy, Marie Brilland Yann. « Riemann hypothesis for the zeta function of a function field over a finite field ». Thesis, Stellenbosch : Stellenbosch University, 2013. http://hdl.handle.net/10019.1/85713.

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Smith, Benjamin Andrew. « Explicit endomorphisms and correspondences ». University of Sydney, 2006. http://hdl.handle.net/2123/1066.

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Doctor of Philosophy (PhD)
In this work, we investigate methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves. Our principal tool is the theory of correspondences, in which homomorphisms of Jacobians are represented by divisors on products of curves. We give families of hyperelliptic curves of genus three, five, six, seven, ten and fifteen whose Jacobians have explicit isogenies (given in terms of correspondences) to other hyperelliptic Jacobians. We describe several families of hyperelliptic curves whose Jacobians have complex or real multiplication; we use correspondences to make the complex and real multiplication explicit, in the form of efficiently computable maps on ideal class representatives. These explicit endomorphisms may be used for efficient integer multiplication on hyperelliptic Jacobians, extending Gallant--Lambert--Vanstone fast multiplication techniques from elliptic curves to higher dimensional Jacobians. We then describe Richelot isogenies for curves of genus two; in contrast to classical treatments of these isogenies, we consider all the Richelot isogenies from a given Jacobian simultaneously. The inter-relationship of Richelot isogenies may be used to deduce information about the endomorphism ring structure of Jacobian surfaces; we conclude with a brief exploration of these techniques.

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Lingenbrink, David Alan Jr. « A New Subgroup Chain for the Finite Affine Group ». Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/hmc_theses/55.

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The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.

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Angulo, Rigo Julian Osorio. « Criptografia de curvas elípticas ». Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/6976.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
According to history, the main objective of cryptography was always to provide security in communications, to keep them out of the reach of unauthorized entities. However, with the advent of the era of computing and telecommunications, applications of encryption expanded to offer security, to the ability to: verify if a message was not altered by a third party, to be able to verify if a user is who claims to be, among others. In this sense, the cryptography of elliptic curves, offers certain advantages over their analog systems, referring to the size of the keys used, which results in the storage capacity of the devices with certain memory limitations. Thus, the objective of this work is to offer the necessary mathematical tools for the understanding of how elliptic curves are used in public key cryptography.
Segundo a história, o objetivo principal da criptografia sempre foi oferecer segurança nas comunicações, para mantê-las fora do alcance de entidades não autorizadas. No entanto, com o advento da era da computação e as telecomunicações, as aplicações da criptografia se expandiram para oferecer além de segurança, a capacidade de: verificar que uma mensagem não tenha sido alterada por um terceiro, poder verificar que um usuário é quem diz ser, entre outras. Neste sentido, a criptografia de curvas elípticas, oferece certas ventagens sobre seu sistemas análogos, referentes ao tamanho das chaves usadas, redundando isso na capacidade de armazenamento dos dispositivos com certas limitações de memória. Assim, o objetivo deste trabalho é fornecer ao leitor as ferramentas matemáticas necessá- rias para a compreensão de como as curvas elípticas são usadas na criptografia de chave pública.

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Reis, Júlio César dos 1979. « Graduações e identidades graduadas para álgebras de matrizes ». [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306363.

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Orientador: Plamen Emilov Kochloukov
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Na presente tese, fornecemos bases das identidades polinomiais graduadas de...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital
Abstract: In this PhD thesis we give bases of the graded polynomial identities of...Note: The complete abstract is available with the full electronic document
Doutorado
Matematica
Doutor em Matemática

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Negreiros, Diogo Bruno Fernandes 1983. « Formas quadráticas, pesos de Hamming generalizados e curvas algébricas ». [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306293.

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Orientador: Paulo Roberto Brumatti
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Este texto tem como objetivo o estudo de um tipo de código que possui relações com as teorias de curvas algébricas e de formas quadráticas. Começaremos introduzindo as definições e resultados sobre as três teorias que serão necessárias a este estudo. Depois apresentaremos os códigos a serem estudados bem como as relações entre seus sub-códigos e curvas algébricas e entre suas palavras e formas quadráticas. Observando que sub-códigos de peso mais baixo correspondem a curvas com mais pontos, nos dedicaremos a obter um processo para a descoberta de sub-códigos de peso mínimo dentro deste tipo de código. Tal processo será possível através de investigações sobre as formas quadráticas associadas a palavras. Finalizaremos com exemplos de aplicações do processo em alguns códigos, o que permite também calcular seus pesos de Hamming generalizados de ordem mais baixa
Abstract: This text's objective is the study of a kind of code wich has relations with the theories of algebraic curves and quadratic forms. We start by introducing definitions and results about the three theories we will need in such study. Later, we present the codes wich will be studied along with relations between its subcodes and algebraic curves and between its words and quadratic forms. Noting that lower weight subcodes correspond to curves with more points, we research a process to find minimum weight subcodes in this kind of code. This process will be possible through investigations on the quadratic forms related to words. Finally we set examples of applications of the process on some codes, and that gives us their lower order generalized Hamming weights
Mestrado
Matematica
Mestre em Matemática

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Marín, Oscar Jhoan Palacio. « Códigos Hermitianos Generalizados ». Universidade Federal de Juiz de Fora (UFJF), 2016. https://repositorio.ufjf.br/jspui/handle/ufjf/2349.

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Nesse trabalho, estamos interessados, especialmente, nas propriedades de duas classes de Códigos Corretores de Erros: os Códigos Hermitianos e os Códigos Hermitianos Generalizados. O primeiro é deﬁnido a partir de lugares do corpo de funções Hermitiano clássico sobre um corpo ﬁnito de ordem quadrada, já o segundo é deﬁnido a partir de uma generalização desse mesmo corpo de funções. Como base para esse estudo, apresentamos ainda resultados da teoria de corpos de funções e outras construções de Códigos Corretores de Erros.
Inthisworkweinvestigatepropertiesoftwoclassesoferror-correctingcodes,theHermitian Codes and their generalization. The Hermitian Codes are deﬁned using the classical Hermitian curve deﬁned over a quadratic ﬁeld. The generalized Hermitian Codes are similar, but uses a generalization of this curve. We also present some results of the theory of function ﬁelds and other constructions of error-correcting codes which are important to understand this work.

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Nardi, Jade. « Quelques retombées de la géométrie des surfaces toriques sur un corps fini sur l'arithmétique et la théorie de l'information ». Thesis, Toulouse 3, 2019. http://www.theses.fr/2019TOU30051.

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Cette thèse, à la frontière entre les mathématiques et l'informatique, est consacrée en partie à l'étude des paramètres et des propriétés des codes de Goppa sur les surfaces de Hirzebruch. D'un point de vue arithmétique, la théorie des codes correcteurs a ravivé la question du nombre de points rationnels d'une variété définie sur un corps fini, qui semblait résolue par la formule de Lefschetz. La distance minimale de codes géométriques donne un majorant du nombre de points rationnels d'une hypersurface d'une variété donnée et de classe de Picard fixée. Ce majorant étant le plus souvent atteint pour les courbes très réductibles, il est naturel de se concentrer sur les courbes irréductibles pour affiner les bornes. On présente une stratégie globale pour majorer le nombre de points d'une variété en fonction de son ambiant et d'invariants géométriques, notamment liés à la théorie de l'intersection. De plus, une méthode de ce type pour les courbes d'une surface torique est développée en adaptant l'idée de F.J Voloch et K.O. Sthör aux variétés toriques. Enfin, on s'intéresse aux protocoles de Private Information Retrivial, qui visent à assurer qu'un utilisateur puisse accéder à une entrée d'une base de données sans révéler d'information sur l'entrée au propriétaire de la base de données. Un protocole basé sur des codes sur des plans projectifs pondérés est proposé ici. Il améliore les protocoles existants en résistant à la collusion de serveurs, au prix d'une grande perte de capacité de stockage. On pallie ce problème grâce à la méthode du lift qui permet la construction de familles de codes asymptotiquement bonnes, avec les mêmes propriétés locales
A part of this thesis, at the interface between Computer Science and Mathematics, is dedicated to the study of the parameters ans properties of Goppa codes over Hirzebruch surfaces. From an arithmetical perspective, the question about number of rational points of a variety defined over a finite field, which seemed dealt with by Lefchetz formula, regained interest thanks to error correcting codes. The minimum distance of an algebraic-geometric codes provides an upper bound of the number of rational points of a hypersurface of a given variety and with a fixed Picard class. Since reducible curves are most likely to reach this bound, one can focus on irreducible curves to get sharper bounds. A global strategy to bound the number of points on a variety depending on its ambient space and some of its geometric invariants is exhibited here. Moreover we develop a method for curves on toric surfaces by adapting F.J. Voloch et K.O. Sthör's idea on toric varieties. Finally, we interest in Private Information Retrivial protocols, which aim to ensure that a user can access an entry of a database without revealing any information on it to the database owner. A PIR protocol based on codes over weighted projective planes is displayed here. It enhances other protocols by offering a resistance to servers collusions, at the expense of a loss of storage capacity. This issue is fixed by a lifting process, which leads to asymptotically good families of codes, with the same local properties

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Sonon, Bernard. « On advanced techniques for generation and discretization of the microstructure of complex heterogeneous materials ». Doctoral thesis, Universite Libre de Bruxelles, 2014. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209087.

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The macroscopic behavior of complex heterogeneous materials is strongly governed by the interactions between their elementary constituents within their microstructure. Beside experimental efforts characterizing the behaviors of such materials, there is growing interest, in view of the increasing computational power available, in building models representing their microstructural systems integrating the elementary behaviors of their constituents and their geometrical organization. While a large number of contributions on this aspect focus on the investigation of advanced physics in material parameter studies using rather simple geometries to represent the spatial organization of heterogeneities, few are dedicated to the exploration of the role of microstructural geometries by means of morphological parameter studies.

The critical ingredients of this second type of investigation are (I) the generation of sets of representative volume elements ( RVE ) describing the geometry of microstructures with a satisfying control on the morphology relevant to the material of interest and (II) the discretization of governing equations of a model representing the investigated physics on those RVEs domains. One possible reason for the under-representation of morphologically detailed RVEs in the related literature may be related to several issues associated with the geometrical complexity of the microstructures of considered materials in both of these steps. Based on this hypothesis, this work is aimed at bringing contributions to advanced techniques for the generation and discretization of microstructures of complex heterogeneous materials, focusing on geometrical issues. In particular, a special emphasis is put on the consistent geometrical representation of RVEs across generation and discretization methodologies and the accommodation of a quantitative control on specific morphological features characterizing the microstructures of the covered materials.

While several promising recent techniques are dedicated to the discretization of arbitrary complex geometries in numerical models, the literature on RVEs generation methodologies does not provide fully satisfying solutions for most of the cases. The general strategy in this work consisted in selecting a promising state-of-the-art discretization method and in designing improved RVE generation techniques with the concern of guaranteeing their seamless collaboration. The chosen discretization technique is a specific variation of the generalized / extended finite element method that accommodates the representation of arbitrary input geometries represented by level set functions. The RVE generation techniques were designed accordingly, using level set functions to define and manipulate the RVEs geometries.

The RVE methodologies developed are mostly morphologically motivated, incorporating governing parameters allowing the reproduction and the quantitative control of specific morphological features of the considered materials. These developments make an intensive use of distance fields and level set functions to handle the geometrical complexity of microstructures. Valuable improvements were brought to the RVE generation methodologies for several materials, namely granular and particle-based materials, coated and cemented geomaterials, polycrystalline materials, cellular materials and textile-based materials. RVEs produced using those developments have allowed extensive testing of the investigated discretization method, using complex microstructures in proof-of-concept studies involving the main ingredients of RVE-based morphological parameter studies of complex heterogeneous materials. In particular, the illustrated approach offers the possibility to address three crucial aspects of those kinds of studies: (I) to easily conduct simulations on a large number of RVEs covering a significant range of morphological variations for a material, (II) to use advanced constituent material behaviors and (III) to discretize large 3D RVEs. Based on those illustrations and the experience gained from their realization, the main strengths and limitations of the considered discretization methods were clearly identified.
Doctorat en Sciences de l'ingénieur
info:eu-repo/semantics/nonPublished

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44

Ethamakula, Bharath Kumar. « Asymptotic Lower Bound for Quasi Transitive Codes over Cubic Finite Fields ». Thesis, 2015. http://etd.iisc.ernet.in/2005/3821.

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Algebraic geometric codes were first introduced by V.D.Goppa . They were well recognized and developed by Tsfasman, Vladut and Zink because they have parameters better than Gilbert-Varshmov bound and thus giving rise to Tsfasman Vladut-Zink bound. While the codes given by Ihara, Tsfasman, Vladut and Zink have complicated construction, Garcia and Stichtenoth on the other hand gave an explicit construction of codes attaining Tsfasman-Vlasut-Zink bound using the terminology of function fields. In coding theory one of the challenging problem is to find a sequence of cyclic codes that are asymptotically good. While this has not been achieved, Stichtenoth generalized cyclic codes to transitive codes and constructed a sequence of asymptotically good transitive codes on algebraic function fields over quadratic finite fields that attain Tsfasman-Vladut-Zink bound. In the case of cubic finite fields, Bezerra, Garcia and Stichtenoth constructed a tower of function fields over cubic finite fields whose limit attains a lower bound and the codes constructed over this tower turns out to be asymptotically good attaining a positive lower bound. Bassa used this tower and constructed quasi transitive codes which are a generalization of transitive codes and proved that they are also asymptotically good and attain the same positive lower bound. Later Bassa, Garcia and Stichtenoth constructed a new tower of function fields over cubic finite fields whose structure is less complicated compared to that of Bezerra, Garcia and Stichtenoths' and proved that codes constructed over it also attain the same positive lower bound. In this work along the lines of Bassa and Stichtenoth we construct quasi transitive codes over the tower given by Bassa, Garcia and Stichtenoth and prove that these quasi transitive codes are also asymptotically good and also attain the same lower bound.

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45

Yanik, Tu��rul. « New methods for finite field arithmetic ». Thesis, 2001. http://hdl.handle.net/1957/32447.

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We describe novel methods for obtaining fast software implementations of the arithmetic operations in the finite field GF(p) and GF(p[superscript k]). In GF(p) we realize an extensive speedup in modular addition and subtraction routines and some small speedup in the modular multiplication routine with an arbitrary prime modulus p which is of arbitrary length. The most important feature of the method is that it avoids bit-level operations which are slow on microprocessors and performs word-level operations which are significantly faster. The proposed method has applications in public-key cryptographic algorithms defined over the finite field GF(p), most notably the elliptic curve digital signature algorithm. The new method provides up to 13% speedup in the execution of the ECDSA algorithm over the field GF(p) for the length of p in the range 161��k��256. In the finite extension field GF(p[superscript k]) we describe two new methods for obtaining fast software implementations of the modular multiplication operation with an arbitrary prime modulus p, which has less bit-length than the word-length of a microprocessor and an arbitrary generator polynomial. The second algorithm is a significant improvement over the first algorithm by using the same concepts introduced in GF(p) arithmetic.
Graduation date: 2002

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46

« Finite fields, algebraic curves and coding theory ». 2006. http://library.cuhk.edu.hk/record=b5896533.

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Yeung Wai Ling Winnie.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.
Includes bibliographical references (leaves 99-100).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.1
Chapter 2 --- Finite Fields --- p.4
Chapter 2.1 --- Basic Properties of Finite Fields --- p.4
Chapter 2.2 --- Existence and Uniqueness of Finite Fields --- p.8
Chapter 2.3 --- Algorithms in Factoring Polynomials --- p.11
Chapter 2.3.1 --- Factorization of xn ´ؤ 1 --- p.11
Chapter 2.3.2 --- Berlekamp Algorithm for Factorizing an Arbitrary Polynomial --- p.13
Chapter 3 --- Algebraic Curves --- p.17
Chapter 3.1 --- Affine and Projective Curves --- p.17
Chapter 3.2 --- Local Properties and Intersections of Curves --- p.19
Chapter 3.3 --- Linear Systems of Curves and Noether's Theorem --- p.24
Chapter 3.4 --- Rational Function and Divisors --- p.29
Chapter 3.5 --- Differentials on a Curve --- p.34
Chapter 3.6 --- Riemann-Roch Theorem --- p.36
Chapter 4 --- Coding Theory --- p.46
Chapter 4.1 --- Introduction to Coding Theory --- p.46
Chapter 4.1.1 --- Basic Definitions for Error-Correcting Code --- p.46
Chapter 4.1.2 --- Geometric Approach to Error-Correcting Capabilities of Codes --- p.48
Chapter 4.2 --- Linear Codes --- p.49
Chapter 4.2.1 --- The Dual of a Linear Code --- p.54
Chapter 4.2.2 --- Syndrome Decoding --- p.57
Chapter 4.2.3 --- Extension of Basic Field --- p.60
Chapter 4.3 --- The Main Problem in Coding Theory --- p.62
Chapter 4.3.1 --- "Elementary Results on Aq(n, d)" --- p.63
Chapter 4.3.2 --- "Lower Bounds on Aq(n, d)" --- p.63
Chapter 4.3.3 --- "Upper Bounds on Aq(n,d)" --- p.65
Chapter 4.3.4 --- Asymptotic Bounds --- p.67
Chapter 4.4 --- Rational Codes --- p.68
Chapter 4.4.1 --- Hamming Codes --- p.68
Chapter 4.4.2 --- Codes on an Oval --- p.69
Chapter 4.4.3 --- Codes on a Twisted Cubic Curve --- p.78
Chapter 4.4.4 --- Normal Rational Codes --- p.82
Chapter 4.5 --- Goppa Codes --- p.84
Chapter 4.5.1 --- Classical Goppa Codes --- p.85
Chapter 4.5.2 --- Geometric Goppa Codes --- p.88
Chapter 4.5.3 --- Good Codes from Algebraic Geometry --- p.91
Chapter 4.6 --- A Recent Non-linear Code Improving the Tsfasman- Vladut-Zink Bound --- p.93
Bibliography --- p.99

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47

« A survey on Calabi-Yau manifolds over finite fields ». 2008. http://library.cuhk.edu.hk/record=b5896863.

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Mak, Kit Ho.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.
Includes bibliographical references (leaves 78-81).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.7
Chapter 2 --- Preliminaries on Number Theory --- p.10
Chapter 2.1 --- Finite Fields --- p.10
Chapter 2.2 --- p-adic Numbers --- p.13
Chapter 2.3 --- The Teichmuller Representatives --- p.16
Chapter 2.4 --- Character Theory --- p.18
Chapter 3 --- Basic Calabi-Yau Geometry --- p.26
Chapter 3.1 --- Definition and Basic Properties of Calabi-Yau Manifolds --- p.26
Chapter 3.2 --- Calabi-Yau Manifolds of Low Dimensions --- p.29
Chapter 3.3 --- Constructions of Calabi-Yau Manifolds --- p.32
Chapter 3.4 --- Importance of Calabi-Yau Manifolds in Physics --- p.35
Chapter 4 --- Number of Points on Calabi-Yau Manifolds over Finite Fields --- p.39
Chapter 4.1 --- The General Method --- p.39
Chapter 4.2 --- The Number of Points on a Family of Calabi-Yau Varieties over Finite Fields --- p.45
Chapter 4.2.1 --- The Case ψ = 0 --- p.45
Chapter 4.2.2 --- The Case ψ ß 0 --- p.50
Chapter 4.3 --- The Number of Points on the Affine Mirrors over Finite Fields --- p.55
Chapter 4.3.1 --- The Case ψ = 0 --- p.55
Chapter 4.3.2 --- The Case ψ ß 0 --- p.56
Chapter 4.4 --- The Number of points on the Projective Mirror over Finite Fields --- p.59
Chapter 4.5 --- Summary of the Results and Related Conjectures --- p.61
Chapter 5 --- The Relation Between Periods and the Number of Points over Finite Fields modulo q --- p.67
Chapter 5.1 --- Periods of Calabi-Yau Manifolds --- p.67
Chapter 5.2 --- The Case for Elliptic Curves --- p.69
Chapter 5.2.1 --- The Periods of Elliptic Curves --- p.69
Chapter 5.2.2 --- The Number of Fg-points on Elliptic Curves Modulo q --- p.70
Chapter 5.3 --- The Case for a Family of Quintic Threefolds --- p.73
Chapter 5.3.1 --- The Periods of Xψ --- p.73
Chapter 5.3.2 --- The Number of F9-points on Quintic Three- folds Modulo q --- p.75
Bibliography --- p.78

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48

« Elliptic curve over finite field and its application to primality testing and factorization ». 1998. http://library.cuhk.edu.hk/record=b5889507.

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by Chiu Chak Lam.
Thesis submitted in: June, 1997.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.
Includes bibliographical references (leaves 67-69).
Abstract also in Chinese.
Chapter 1 --- Basic Knowledge of Elliptic Curve --- p.2
Chapter 1.1 --- Elliptic Curve Group Law --- p.2
Chapter 1.2 --- Discriminant and j-invariant --- p.7
Chapter 1.3 --- Elliptic Curve over C --- p.10
Chapter 1.4 --- Complex Multiplication --- p.15
Chapter 2 --- Order of Elliptic Curve Group Over Finite Fields and the Endo- morphism Ring --- p.18
Chapter 2.1 --- Hasse's Theorem --- p.18
Chapter 2.2 --- The Torsion Group --- p.23
Chapter 2.3 --- The Weil Conjectures --- p.33
Chapter 3 --- Computing the Order of an Elliptic Curve over a Finite Field --- p.35
Chapter 3.1 --- Schoof's Algorithm --- p.35
Chapter 3.2 --- Computation Formula --- p.38
Chapter 3.3 --- Recent Works --- p.42
Chapter 4 --- Primality Test Using Elliptic Curve --- p.43
Chapter 4.1 --- Goldwasser-Kilian Test --- p.43
Chapter 4.2 --- Atkin's Test --- p.44
Chapter 4.3 --- Binary Quadratic Form --- p.49
Chapter 4.4 --- Practical Consideration --- p.51
Chapter 5 --- Elliptic Curve Factorization Method --- p.54
Chapter 5.1 --- Lenstra's method --- p.54
Chapter 5.2 --- Worked Example --- p.56
Chapter 5.3 --- Practical Considerations --- p.56
Chapter 6 --- Elliptic Curve Public Key Cryptosystem --- p.59
Chapter 6.1 --- Outline of the Cryptosystem --- p.59
Chapter 6.2 --- Index Calculus Method --- p.61
Chapter 6.3 --- Weil Pairing Attack --- p.63

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49

(11199984), Frankie Chan. « Finite quotients of triangle groups ». Thesis, 2021.

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Extending an explicit result from Bridson–Conder–Reid, this work provides an algorithm for distinguishing finite quotients between cocompact triangle groups Δ ?and lattices Γ of constant curvature symmetric 2-spaces. Much of our attention will be on when these lattices are Fuchsian groups. We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(n,F_q) (for an odd prime power q), or an Abelian extension of one of these 3 groups. For the latter case, we will require and develop an approach for creating group extensions upon a shared finite quotient of Δ? and Γ which between them have differing degrees of smoothness. Furthermore, on the order of a finite quotient that distinguishes between ?Δ and Γ, we are able to establish an effective upperbound that is superexponential depending on the cone orders appearing in each group.

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50

Videla, Guzman Denis Eduardo. « El espectro de códigos cíclicos y grafos asociados ». Doctoral thesis, 2018. http://hdl.handle.net/11086/6602.

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Una de las clases más importantes e implementadas de códigos, es la clase de los códigos cíclicos, debido a su eficiente codificación, y por la existencia de buenos algoritmos para decodificarlos. Por otro lado, entender la distribución de pesos de códigos permite en algunos casos, calcular el error de probabilidad a la hora de decodificar. Por ello, es importante conocer la distribución de pesos de códigos cíclicos. En general, el problema de calcular distribuciones de pesos es computacionalmente complejo, inclusive en el caso de códigos cíclicos. Sin embargo, es posible atacar este problema si pedimos ciertas condiciones al código cíclico. Esta tesis se centra en el estudio del espectro o distribución de pesos de códigos cíclicos, y de las distintas relaciones que tienen estos espectros con otros objetos que aparecen en el estudio de cuerpos finitos tales como sumas exponenciales, caracteres, curvas algebraicas y grafos de Cayley.

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Thèses sur le sujet « Finite fields (Algebra) Geometry »

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