A New Implementation Of Miura-Arita Algorithm For Miura Curves

Lecture Notes in Computer Science, 2010

Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups H 1 (Gal(k/k), E) → H 1 (Gal(k/k), A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

Journal de Théorie des Nombres de Bordeaux, 2018

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2, 3, 4, 5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for minimising bidegree (2, 2)forms, 3 × 3 × 3 cubes and 2 × 2 × 2 × 2 hypercubes. We also prove a theorem relating the minimal discriminant to that of the Jacobian elliptic curve.

Bulletin of the American Mathematical Society, 1986

BARRY MAZUR Bibliography 1. General references 2. References requiring some background 3. Accounts of the proof of Mordell's conjecture which require familiarity with the techniques of number theory or algebraic geometry 4. Other works cited (given in alphabetical order) I. WHAT ARE DIOPHANTINE PROBLEMS? I don't know the answer. To feel our way around the question, let us first consider some miscellaneous problems that have been labelled " Diophantine" and then review some attempts towards a systematic organization of these "Diophantine problems". The point of such a preamble is to get us to appreciate why number-theorists are often so fondly devoted to the study of rational points on algebraic curves, and to sense the more general contexts which embrace that study. My aim will then be to discuss the recent progress made in the arithmetic of curves [Fa]', 1 and to explain a few of the ideas involved without requiring substantial background in algebra, number theory, or algebraic geometry. This survey has been culled from my notes to the Colloquium Series Lectures at the 1984 winter meeting of the American Mathematical Society at Louisville, Kentucky, and from part of the Albert Lectures delivered at the University of Chicago this past fall. I feel very lucky to have had such engaging and stimulating audiences. I am very appreciative of the comments and suggestions of J.-P. Serre who read closely early versions of this survey. I am also thankful for the help and advice I received from many people in the course of writing it, among whom are:

Mathematics of Computation, 1990

This volume has been compiled in honour of the well known mathematician Hans Zasscnhaus on the occasion of his 75th birthday. As colleagues, collaborators and friends, we dedicate this work to him in the hope that it might inspire present and future researchers, in a similar fashion to the way in which his brilliant ideas filled us with the creati>e urge. Hans Zassenhaus was born in Koblenz (West Germany) on 28 May 1912 and brought up in Hamburg. There he studied mathematics under the supervision of E. Artin, E. Heeke, and E. Sperner. and was also a student of physics and biology. He was awarded his PhD at the early age of 22 with a thesis on "Kennzeichnung linearer Gruppen als Permutationsgruppen••. In the subsequent two years, which he spent at Rostock as a teaching assistant, he wrote his famous monograph on group theory which is still among th..: standard textbooks on that subject. In 1938-back in Hamburg-he qualified for a full teaching appointment with a paper on Lie rings of prime characteristic. In 1946 he was appointed associate professor and director of the Institute for Applied \1athcmatics which he had founded at Hamburg University in the same year. Accepting the challenge of an offer to help McGill University in the building up of Canadian graduate education in mathematics he left his country in 1949 for Montreal, Canada, where he was later joined by his wife and children. As Peter Redpath Professor at McGill University he supervised the PhD studies of many Canadian students. In 1957 he became a Canadian citizen. In 1959 he moved to the USA where he taught at Notre Dame (1959-1963) and at Ohio State University in Columbus. At OSU he hdd the position of research professor until his retirement five years ago. During these years he frequently visited other universities as a guest professor. We briefly mention the academic year 1955-56 at Princeton, two years at the California Institute of Technology as a Fairchild Distinguished Scholar, a Gauss professorship at Gottingen in 1967 and the US Senior Scientist Award of the Humboldt-Stiftung. In 1956 he became a Fellow of the Royal Canadian Society and in 1969 Editor-in-Chief of the Journal of Number Theory. Among the mathematicians of our time he is one of the few still active in different areas-we have already mentioned his contributions to group theory. Most graduate students learn his famous "butterfly lemma" which nowadays forms a substantial part of the proof of the Jordan-Holder-Schreier theorem. In 1978 (jointly with R. Bulow, J. Neubiiser and H. Wondraschek) he wrote a book on crystallographic groups which seems to be better known to physicists than to the mathematical community. Orders (and their ideal theory) are also among the central objectives of his research. The constructive approach clearly dominates. In a joint paper with E. C. Dade and 0. Taussky it was proven that the (n-1)st power of a fractional ideal of an order of rank n over the integers is always invertible. The authors obtained the idea of that theorem by numerical calculations on a computer and then each of them gave a different proof. This was an early and powerful demonstration of the usefulness of mathematical experimenting by computer. Later on he developed several algorithms for the embedding of an order into its maximal order. Each algorithm improved the preceding one and the numerical results obtained by each algorithm led to further theoretical improvements-0747-7171"87.040001

Revista Colombiana De Matematicas, 2013

We give a simple and effective method for the construction of algebraic curves over finite fields with many rational points. The curves are given as Kummer covers of the projective line.

This paper presents the derivation of a new algorithm for multiplying of two Kaluza numbers. Performing this operation directly requires 1024 real multiplications and 992 real additions. The proposed algorithm can compute the same result with only 512 real multiplications and 576 real additions. The derivation of our algorithm is based on utilizing the fact that multiplication of two Kaluza numbers can be expressed as a matrixvector product. The matrix multiplicand that participates in the product calculating has unique structural properties. Namely exploitation of these specific properties leads to significant reducing of the complexity of Kaluza numbers multiplication.

Bulletin of the Brazilian Mathematical Society, New Series, 2020

A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r 2 − s 2 , where θ ∈ (0, π), cos(θ) = s/r , and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235-241, 1997) that N is a θ-congruent number if and only if the elliptic curve E θ N : y 2 = x(x + (r + s)N)(x − (r − s)N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N = 1, 2, 3, 6 is a θ-congruent number if and only if rank of E θ N (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ-triangle for a θ-congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E θ N (Q). Then, based on the addition of two distinct points in E θ N (Q), we provide a way to find new rational θ-triangles for the θ-congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E θ N (Q) with corresponding rational θ-triangles.

IEEE Transactions on Computers, 2000

Hyperelliptic curves (HEC) look promising for cryptographic applications, because of their short operand size compared to other public-key schemes. The operand sizes seem well suited for small processor architectures, where memory and speed are constrained. However, the group operation has been believed to be too complex and, thus, HEC have not been used in this context so far. In recent years, a lot of effort has been made to speed up group operation of genus-2 HEC. In this paper, we increase the efficiency of the genus-2 and genus-3 hyperelliptic curve cryptosystems (HECC). For certain genus-3 curves, we can gain almost 80 percent performance for a group doubling. This work not only improves Gaudry and Harley's algorithm [1], but also improves the original algorithm introduced by Cantor [2]. Contrary to common belief, we show that it is also practical for certain curves to use Cantor's algorithm to obtain the highest efficiency for the group operation. In addition, we introduce a general reduction method for polynomials according to Karatsuba. We implemented our most efficient group operations on Pentium and ARM microprocessors.

An attack is possible upon all three RSA analogue PKCs based on singular cubic curves given by Koyama. While saying so, Seng et al observed that the scheme become insecure if a linear relation is known between two plaintexts. In this case, attacker has to compute greatest common divisor of two polynomials corresponding to those two plaintexts. However, the computation of greatest common divisor of two polynomials is not efficient. For the reason, the degree e of both polynomials, an encryption exponent, is quite large.

2007

In the present paper, we investigate discretization of curves based on polynomials in the 2-dimensional space. Under some assumptions, we propose an arithmetic characterization of thin and connected discrete approxima- tions of such curves. In fact, we reach usual discretization models, that is, GIQ, OBQ and BBQ but with a generic arithmetic definition.