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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
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The Calculus of Signal Flow Diagrams I
TL;DR: A graphical syntax for signal flow diagrams based on the language of symmetric monoidal categories is introduced and it is shown that any diagram of this syntax can be realised, via rewriting in the equational theory, as a signal flow graph.
Book ChapterDOI
Improving Group Law Algorithms for Jacobians of Hyperelliptic Curves
TL;DR: In this paper, three ideas to speed up the computation of the group operation in the Jacobian of a hyperelliptic curve are proposed.
Posted Content
Efficient Quantum Algorithms for Estimating Gauss Sums
Wim van Dam,Gadiel Seroussi +1 more
TL;DR: An efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings is presented and evidence is given that this problem is hard for classical algorithms.
Journal ArticleDOI
Commensurability classes of hyperbolic Coxeter groups
TL;DR: In this paper, the hyperbolic Coxeter n-simplex reflection groups up to widecommensurability for all n 3 were classified up to a wide range of subgroups.
Journal ArticleDOI
Computing class fields via the Artin map
TL;DR: Based on an explicit representation of the Artin map for Kummer extensions, a method to compute arbitrary class fields in the case where the field contains sufficiently many roots of unity is presented.
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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