Open AccessBook
A Course in Computational Algebraic Number Theory
Reads0
Chats0
TLDR
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.Abstract:
A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.read more
Citations
More filters
Journal ArticleDOI
Elliptic curves associated with simplest quartic fields
TL;DR: In this article, the authors studied the infinite family of elliptic curves associated with simplest cubic fields and obtained the structure of the Mordell-Weil group and all integral points on the original model of the curve.
Dissertation
Évaluation efficace de fonctions numériques - Outils et exemples
TL;DR: Les systemes informatiques permettent d'evaluer des fonctions numeriques telles que f = exp, sin, arccos, etc s'interesse au processus d'implementation de ces fonications, mais l'objectif est toujours d'obtenir l'im implementation la plus efficace possible.
Journal ArticleDOI
Square form factorization
TL;DR: A detailed analysis of SQUFOF, Daniel Shanks' Square Form Factorization algorithm, giving the average time and space requirements and the effect of multipliers used for a single factorization or when racing the algorithm in parallel.
Book ChapterDOI
Computing CM points on shimura curves arising from cocompact arithmetic triangle groups
TL;DR: The Shimura reciprocity law is algorithmically applied to compute CM points $j(z_D) \in {\mathbb P}^1_{\mathbb C}$ and their Galois conjugates so as to recognize them as purported algebraic numbers.
Book ChapterDOI
Fast Multi-scalar Multiplication Methods on Elliptic Curves with Precomputation Strategy Using Montgomery Trick
Katsuyuki Okeya,Kouichi Sakurai +1 more
TL;DR: The Montgomery trick is applied to reduce the number of inversions required with a width window w from O(22w) to O(w), which enables the removal of points that will not be used later from the process of precomputation.
References
More filters
Book
A Course of Modern Analysis
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Journal ArticleDOI
Modular multiplication without trial division
TL;DR: A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.
Book
Advanced Topics in the Arithmetic of Elliptic Curves
TL;DR: In this article, the authors continue the study of elliptic curves by presenting six important, but somewhat more specialized topics: Elliptic and modular functions for the full modular group.
Journal ArticleDOI
Improved methods for calculating vectors of short length in a lattice, including a complexity analysis
U. Fincke,Michael Pohst +1 more
TL;DR: In this paper, the authors show that searching through an ellipsoid is in many cases much more efficient than enumerating all vectors of Z'.. in a suitable box.
Journal ArticleDOI
Lattice basis reduction: improved practical algorithms and solving subset sum problems
Claus-Peter Schnorr,M. Euchner +1 more
TL;DR: Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
Related Papers (5)
A method for obtaining digital signatures and public-key cryptosystems
Graduate Texts in Mathematics
Rajendra Bhatia,Glen Bredon,Wolfgang Walter,Joseph J. Rotman,M. Ram Murty,Jane Gilman,Peter Walters,Martin Golubitsky,Ioannis Karatzas,Henri Cohen,Raoul Bott,Gaisi Takeuti,Béla Bollobás,John M. Lee,Jiří Matoušek,Saunders Mac Lane,John L. Kelley,B. A. Dubrovin,Tom M. Apostol,John Stillwell,William Arveson +20 more