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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
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Journal ArticleDOI
Quartic rings associated to binary quartic forms
TL;DR: In this paper, a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring was given, and a geometric interpretation of this parametrization was given.
Journal ArticleDOI
Valuations of p-adic regulators of cyclic cubic fields
Tommy Hofmann,Yinan Zhang +1 more
TL;DR: In this article, the p-adic regulator of cyclic cubic extensions of Q with discriminant up to 1016 for 3 p 100, and the distribution of the padic valuation of the regulators are observed.
Posted Content
The $p$-adic CM-method for genus 2
Book ChapterDOI
Parallel Lattice Basis Reduction Using a Multi-threaded Schnorr-Euchner LLL Algorithm
Werner Backes,Susanne Wetzel +1 more
TL;DR: This new, multi-threaded algorithm is the first to provide an efficient, parallel implementation of the Schorr-Euchner algorithm for today's multi-processor,Multi-core computer architectures.
Book ChapterDOI
Efficient Implementation of Cryptosystems Based on Non-maximal Imaginary Quadratic Orders
TL;DR: This work introduces an efficient batch decryption method for NICE, which allows to speed up the decryption by about 30% for a batch size of 100 messages and introduces an entirely new arithmetic for elements in Ker(ΦCl-1), which uses the generator and ring-equivalence for exponentiation.
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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