Open AccessBook
A Course in Computational Algebraic Number Theory
Henri Cohen
- 01 Jan 1993
3.2K
TL;DR: 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.
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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.
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Citations
Fast computation of Hermite normal forms of random integer matrices
Clément Pernet,William Stein +1 more
TL;DR: This paper describes the fastest implementation for computing Hermite normal form for large matrices with large entries, and proposes significant improvements to the algorithm by Micciancio and Warinschi and extends these techniques to the computation of the saturation of a matrix.
36
On quadratic fields with large 3-rank
TL;DR: It is proved that Q( √ −5393946914743) is the smallest imaginary quadratic field with 3-rank equal to 5, and a simpler proof of Davenport and Heilbronn's theorem is given.
Cuts from proofs: a complete and practical technique for solving linear inequalities over integers
Isil Dillig,Thomas Dillig,Alex Aiken +2 more
- 01 Dec 2011
TL;DR: This work proposes a novel, sound, and complete Simplex-based algorithm, which can be viewed as a semantic generalization of the branch-and-bound technique, that systematically discovers and excludes entire subspaces of the solution space containing no integer points.
36
Moduli Spaces for Rings and Ideals
Melanie Matchett Wood
- 01 Jan 2009
TL;DR: In this article, it was shown that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary cubic forms, and the correspondence between forms and algebraic data are functorial in the base scheme.
36
Classification of eight dimensional perfect forms
TL;DR: In this article, the authors classify the perfect lattices in dimension 8 using symmetry in polyhedral computations and describe algorithms making the classification possible, which heavily relies on exploiting symmetry.
References
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
Joseph H. Silverman
- 01 Jan 1994
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.
2.2K
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.
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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