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A Course in Computational Algebraic Number Theory
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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
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Journal ArticleDOI
Finite element-based algorithms to make cuts for magnetic scalar potentials: topological constraints and computational complexity
Paul W. Gross,P.R. Kotiuga +1 more
TL;DR: This paper outlines a generic algorithm to generate cuts for magnetic scalar potentials in 3-dimensional multiply-connected finite element meshes based on the algebraic structures of (co)homology theory with differential forms and developed in the context of the finite element method and finite element data structures.
Dissertation
Descents on curves of Genus 1
TL;DR: This thesis improves on various methods connected with computing the Mordell-Weil group of an elliptic curve and gives a new upper bound for the difference of the logarithmic and canonical heights of points on elliptic curves.
Journal ArticleDOI
Lattices of compatibly embedded finite fields
TL;DR: A coherent scheme for solving this problem based on an efficient method for compatibly embedding one field within another is presented, which forms a central component of the MAGMA module for finite fields.
Book ChapterDOI
Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences
Jingguo Bi,Jean-Sébastien Coron,Jean-Charles Faugère,Phong Q. Nguyen,Guénaël Renault,Rina Zeitoun +5 more
TL;DR: The first significant speedups over Coppersmith's algorithm are presented, based on a special property of the matrices used by Copperssmith's algorithm, which allows us to provably speed up the LLL reduction by rounding, and which can also be used to improve the complexity analysis of Coppermith's original algorithm.
Journal ArticleDOI
HT90 and “simplest” number fields
TL;DR: In this article, a condition (M) was proposed for proving the existence of polynomials whose zeros satisfy (1.1) for a given number field of degrees from 3 to 6.
References
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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.
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