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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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Proceedings Article
Finding ECM-friendly curves through a study of Galois properties
TL;DR: In this paper, the authors prove some divisibility properties of elliptic curves modulo primes and explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization.
Book ChapterDOI
Reducing Logarithms in Totally Non-maximal Imaginary Quadratic Orders to Logarithms in Finite Fields
Detlef Hühnlein,Tsuyoshi Takagi +1 more
TL;DR: The recently proposed efficient DSA-analogue in totally non-maximal imaginary quadratic order O Δp are only as secure as the original DSA scheme based on finite fields and hence loose much of its attractiveness.
Book ChapterDOI
Inversion of Circulant Matrices over Zm
TL;DR: The algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over Z m, and three different algorithms which do not use this approach are presented.
Book ChapterDOI
New Families of ECM Curves for Cunningham Numbers
Eric Brier,Christophe Clavier +1 more
TL;DR: The aim is to build families of elliptic curves more efficient to help factoring numbers of special form, including numbers from the Cunningham Project, by classify and give a parameterization of modular curves for some torsion subgroups.
Posted Content
A complexity dichotomy for hypergraph partition functions
TL;DR: A dichotomy theorem is given for r > 2, showing for which H this problem is in FP and forWhich H it is #P-complete, which generalizes a theorem of Dyer and Greenhill (2000) for the case r = 2, which corresponds to counting graph homomorphisms.
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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