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
Periodic Continued Fractions in Elliptic Function Fields
Alfred J. van der Poorten,Xuan Chuong Tran +1 more
- 07 Jul 2002
TL;DR: All families of quartic polynomials over Q whose square root has a periodic continued fraction expansion, and detail those expansions, prove that, contrary to expectation, the cases of period length nine and eleven do not occur.
19
•Dissertation
Primality Tests on Commutator Curves
Sebastian Wedeniwski
- 01 Jan 2001
Abstract: 3 2 ln(n) 2 durchgeführt werden muss. Die drei Appendizes haben den folgenden Inhalt: (1) Appendix A gibt eine obere Schranke für die kleinsten quadratischen Nichtreste an. Dies findet in Kapitel 6 Verwendung. (2) Appendix B enthält Tabellen, die als Basis für Beobachtungen und Vermutungen in dieser Arbeit allgemein dienen. (3) Appendix C bietet fünf unterschiedliche Implementierungen und Laufzeitver-gleiche meines Hypothetical Commutator Curve Primality Test an.
19
HT90 and “simplest” number fields
TL;DR: In this article, a condition (M) was proposed for proving the existence of polynomials whose zeros satisfy (1.1) for a given number field of degrees from 3 to 6.
LLL reducing with the most significant bits
Saruchi,Ivan Morel,Damien Stehlé,Gilles Villard +3 more
- 23 Jul 2014
TL;DR: A sufficient condition on the closeness between B and B is given so that an LLL-reducing transformation U for B remains valid for B, and an efficient reduction algorithm when B is itself a small deformation of an L LL-reduced basis is analyzed.
On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic
Ichiro Shimada,Tetsuji Shioda +1 more
TL;DR: In this paper, the authors give a defining equation for a complex Fermat quartic surface with 56 lines, which is isomorphic to the complex smooth quadratic surface with only 48 lines.
19
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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