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A Course in Computational Algebraic Number Theory
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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
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Modular square root puzzles: Design of non-parallelizable and non-interactive client puzzles
Yves Igor Jerschow,Martin Mauve +1 more
TL;DR: A novel scheme for client puzzles which relies on the computation of square roots modulo a prime and is able to mitigate DoS attacks on hosts in 1 or even 10 Gbit networks is introduced.
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Velocity and velocity bounds in static spherically symmetric metrics
TL;DR: In this article, the velocity of massless particles with dependence on the distance, r, in Schwarzschild coordinates is derived for the case of Reissner-Nordstrom with and without the cosmological constant.
Journal ArticleDOI
The Poincaré Half-Plane for Informationally-Complete POVMs
TL;DR: In this paper, the structure of IC-POVMs is found to be intimately related to the Kochen-Specker theorem, where the congruence (or non-congruence) subgroups of index d of the modular group are translated into groups of permutation gates, some of which are the sought fiducials.
Journal ArticleDOI
Computing Generators of Free Modules over Orders in Group Algebras
Werner Bley,Henri Johnston +1 more
TL;DR: In this article, the Wedderburn decomposition of a number field and a group algebra is explicitly computable and each M χ is in fact a matrix ring over a field, which leads to a necessary and sufficient condition for X to be free of given rank d over A.
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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