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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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Book ChapterDOI
Comparing the difficulty of factorization and discrete logarithm: a 240-digit experiment
Fabrice Boudot,Pierrick Gaudry,Aurore Guillevic,Nadia Heninger,Emmanuel Thomé,Paul Zimmermann +5 more
TL;DR: In this paper, the RSA-240 and RSA-250 factorizations over a 795-bit prime field were reported, showing that computing a discrete logarithm is not much harder than a factorization of the same size.
Journal ArticleDOI
Inversion of circulant matrices over Z m
TL;DR: In this paper, the problem of inverting an n A-n circulant matrix with entries over Zm was considered, and three different algorithms were presented for this problem.
Journal ArticleDOI
Analysis of the Sliding Window Powering Algorithm
TL;DR: This work analyzes precisely the parameters entering the sliding window powering method, and applies this to the case of an elliptic curve over a large finite prime field.
Journal ArticleDOI
Combining Problem Structure with Basis Reduction to Solve a Class of Hard Integer Programs
TL;DR: It is shown that an integer basis of L can be obtained by taking the Kronecker product of vectors frominteger bases of two much smaller lattices, and the resulting basis is a reduced basis if the integer bases of the two small lattices are reduced bases and a suitable ordering is chosen.
Proceedings ArticleDOI
The Arithmetic of Discretized Rotations
TL;DR: This work considers the problem of planar rotation by an irrational angle, where the space is discretized to a lattice by means of a round‐off procedure which preserves invertibility, and admits an embedding into a dynamical system which is expanding with respect to a non‐archimedean metric.
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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