Families of special Weierstrass points

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

Limit weierstrass points on nodal reducible curves

In the 1980s D. Eisenbud and J. Harris posed the following question: What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type? In the present article, we give a partial answer to this question. We consider the case where the limit curve has components intersecting at points in general position and where the degeneration occurs along a general direction. For this case we compute the limits of Weierstrass points of any order. However, for the usual Weierstrass points, of order one, we need to suppose that all of the components of the limit curve intersect each other.

Weierstrass Points with First Non-Gap Four on a Double Covering of a Hyperelliptic Curve

2000 Mathematics Subject Classification: Primary 14H55; Secondary 14H30, 14H40, 20M14.

Corrigendum for "Weierstrass Points with first Non-Gap four on a Double Covering of a Hyperelliptic Curve"

In the proof of Lemma 3.1 in [1] we need to show that we may take the two points p and q with p ≠ q such that p+q+(b-2)g21(C′)∼2(q1+… +qb-1) where q1,…,qb-1 are points of C′, but in the paper [1] we did not show that p ≠ q. Moreover, we hadn't been able to prove this using the method of our paper [1]. So we must add some more assumption to Lemma 3.1 and rewrite the statements of our paper after Lemma 3.1. The following is the correct version of Lemma 3.1 in [1] with its proof.

Weierstrass Points with First Non-Gap Four on a Double Covering of a Hyperelliptic Curve II

2000 Mathematics Subject Classification: Primary 14H55; Secondary 14H30, 14J26.

Surfaces de Riemann compactes et formule de trace d'Eichler

Dans ce mémoire, nous étudierons quelques propriétés algébriques, géométriques et topologiques des surfaces de Riemann compactes. Deux grand sujets seront traités. Tout d'abord, en utilisant le fait que toute surface de Riemann compacte de genre g plus grand ou égal à 2 possède un nombre fini de points de Weierstrass, nous allons pouvoir conclure que ces surfaces possèdent un nombre fini d'automorphismes. Ensuite, nous allons étudier de plus près la formule de trace d'Eichler. Ce théorème nous permet de trouver le caractère d'un automorphisme agissant sur l'espace des q-différentielles holomorphes. Nous commencerons notre étude en utilisant la quartique de Klein. Nous effectuerons un exemple de calcul utilisant le théorème d'Eichler, ce qui nous permettra de nous familiariser avec l'énoncé du théorème. Finalement, nous allons démontrer la formule de trace d'Eichler, en prenant soin de traiter le cas où l'automorphisme agit sans point fixe séparément du cas où l'automorphisme possède des points fixes.

On certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties.

Eventually minimal curves

A curve defined over a finite field is maximal or minimal according to whether the number of rational points attains the upper or the lower bound in Hasse-Weil's theorem, respectively. In the study of maximal curves a fundamental role is played by an invariant linear system introduced by Ruck and Stichtenoth in [6]. In this paper we define an analogous invariant system for minimal curves, and we compute its orders and its Weierstrass points. In the last section we treat the case of curves having genus three in characteristic two.

Teorema de Riemann-Roch e aplicações

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

Jacobi quartic curves revisited

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.

Faster pairings on special Weierstrass curves

This paper presents efficient formulas for computing cryptographic pairings on the curve y 2 = c x 3 + 1 over fields of large characteristic. We provide examples of pairing-friendly elliptic curves of this form which are of interest for efficient pairing implementations.

Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.

Communications and learning structures that facilitate transfer of knowledge at innovation transition points

Innovation processes are rarely smooth and disruptions often occur at transition points were one knowledge domain passes the technology on to another domain. At these transition points communication is a key component in assisting the smooth hand over of technologies. However for smooth transitions to occur we argue that appropriate structures have to be in place and boundary spanning activities need to be facilitated. This paper presents three case studies of innovation processes and the findings support the view that structures and boundary spanning are essential for smooth transitions. We have explained the need to pass primary responsibility between agents to successfully bring an innovation to market. We have also shown the need to combine knowledge through effective communication so that absorptive capacity is built in process throughout the organisation rather than in one or two key individuals.

Points of convergence and divergence : normative, institutional and operational relationships between R2P and POC

As an international norm, the Responsibility to Protect (R2P) has gained substantial influence and institutional presence—and created no small controversy—in the ten years since its first conceptualisation. Conversely, the Protection of Civilians in Armed Conflict (PoC) has a longer pedigree and enjoys a less contested reputation. Yet UN Security Council action in Libya in 2011 has thrown into sharp relief the relationship between the two. UN Security Council Resolutions 1970 and 1973 follow exactly the process envisaged by R2P in response to imminent atrocity crimes, yet the operative paragraphs of the resolutions themselves invoke only PoC. This article argues that, while the agendas of PoC and R2P converge with respect to Security Council action in cases like Libya, outside this narrow context it is important to keep the two norms distinct. Peacekeepers, humanitarian actors, international lawyers, individual states and regional organisations are required to act differently with respect to the separate agendas and contexts covered by R2P and PoC. While overlap between the two does occur in highly visible cases like Libya, neither R2P nor PoC collapses normatively, institutionally or operationally into the other.