We classify smooth complex projective algebraic curves C of low genus 7\leq g\leq 10 such that the variety of nets W_{g-1}^{2}(C) has dimension g- 7 . We show that \dim W_{g-1}^{2}(C)=g- 7 is ...equivalent to the following conditions according to the values of the genus g . (\mathrm{i})C is either trigonal, a double covering of a curve of genus 2 or a smooth plane curve degree 6 for g=10 . (\mathrm{ii})C is either trigonal, a double covering of a curve of genus 2, a tetragonal curve with a smooth model of degree 8 in P^{3} or a tetragonal curve with a plane model of degree 6 for g=9 . (\mathrm{iii})C is either trigonal or has a birationally very ample g_{6}^{2} for g=8 or g=7 .
Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding ...theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences.
QUADRIC, CUBIC AND QUARTIC CONES KORCHAGIN, ANATOLY B.; WEINBERG, DAVID A.
The Rocky Mountain journal of mathematics,
01/2005, Volume:
35, Issue:
5
Journal Article
Peer reviewed
Open access
There are 2 irreducible quadric cones (real and imaginary) required for obtaining the affine classification of the 4 irreducible conic sections. According to Newton there are 5 irreducible cubic ...cones required for obtaining his classification of 59 irreducible cubic sections. In this historical survey paper we show how it follows from Gudkov's classification of forms of real projective quartic curves that 1037 quartic cones are required for obtaining a similar classification of irreducible quartic sections. We also present the singular-isotopy classification of the unions of irreducible affine cubic curves with their asymptotes, which consists of 99 classes. This classification sheds a new light on Newton's famous classification consisting of 78 species.
Finding points on curves over finite fields VON ZUR GATHEN, Joachim; SHPARLINSKI, Igor; SINCLAIR, Alistair
SIAM journal on computing,
01/2003, Volume:
32, Issue:
6
Journal Article
Peer reviewed
We solve two computational problems concerning plane algebraic curves over finite fields: generating a uniformly random point, and finding all points deterministically in amortized polynomial time ...(over a prime field, for nonexceptional curves).
In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether ...expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at P together with functions for each value in this semigroup, provided P is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.
L'objet de cette thèse est l'étude de diverses primitives cryptographiques utiles dans des protocoles Diffie-Hellman. Nous étudions tout d'abord les protocoles Diffie-Helmman sur des structures ...commutatives ou non. Nous en proposons une formulation unifiée et mettons en évidence les différents problèmes difficiles associés dans les deux contextes. La première partie est consacrée à l'étude de pseudo-paramétrisations de courbes algébriques en temps constant déterministe, avec application aux fonctions de hachage vers les courbes. Les propriétés des courbes algébriques en font une structure de choix pour l'instanciation de protocoles reposant sur le problème Diffie-Hellman. En particulier, ces protocoles utilisent des fonctions qui hachent directement un message vers la courbe. Nous proposons de nouvelles fonctions d'encodage vers les courbes elliptiques et pour de larges classes de fonctions hyperelliptiques. Nous montrons ensuite comment l'étude de la géométrie des tangentes aux points d'inflexion des courbes elliptiques permet d'unifier les fonctions proposées tant dans la littérature que dans cette thèse. Dans la troisième partie, nous nous intéressons à une nouvelle instanciation de l'échange Diffie-Hellman. Elle repose sur la difficulté de résoudre un problème de factorisation dans un anneau de polynômes non-commutatifs. Nous montrons comment un problème de décomposition Diffie-Hellman sur un groupe non-commutatif peut se ramener à un simple problème d'algèbre linéaire pourvu que les éléments du groupe admettent une représentation par des matrices. Bien qu'elle ne soit pas applicable directement au cas des polynômes tordus puisqu'ils n'ont pas d'inverse, nous profitons de l'existence d'une notion de divisibilité pour contourner cette difficulté. Finalement, nous montrons qu'il est possible de résoudre le problème Diffie-Hellman sur les polynômes tordus avec complexité polynomiale.
In this thesis, we study several cryptographic primitives of use in Diffie-Hellman like protocols. We first study Diffie-Hellman protocols on commutative or noncommutative structures. We propose an unified wording of such protocols and bring out on which supposedly hard problem both constructions rely on. The first part is devoted to the study of pseudo-parameterization of algebraic curves in deterministic constant time, with application to hash function into curves. Algebraic curves are indeed particularly interesting for Diffie-Hellman like protocols. These protocols often use hash functions which directly hash into the curve. We propose new encoding functions toward elliptic curves and toward large classes of hyperelliptic curves. We then show how the study of the geometry of flex tangent of elliptic curves unifies the encoding functions as proposed in the litterature and in this thesis. In the third part, we are interested in a new instantiation of the Diffie-Hellman key exchange. It relies on the difficulty of factoring in a non-commutative polynomial ring. We show how to reduce a Diffie-Hellman decomposition problem over a noncommutative group to a simple linear algebra problem, provided that group elements can be represented by matrices. Although this is not directly relevant to the skew polynomial ring because they have no inverse, we use the divisibility to circumvent this difficulty. Finally, we show it's possible to solve the Diffie-Hellman problem on skew polynomials with polynomial complexity.
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