Open AccessBook
A Course in Computational Algebraic Number Theory
Reads0
Chats0
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
More filters
Journal ArticleDOI
A general Lagrange Theorem
TL;DR: It may be useful to have at hand a simple and virtually computation-free proof of a general Lagrange Theorem for the ordinary continued fractions expansion of a real number.
Proceedings ArticleDOI
A linear time algorithm for quantum 2-SAT
Niel de Beaudrap,Sevag Gharibian +1 more
TL;DR: This paper presents an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively.
Proceedings ArticleDOI
Solvability by radicals from an algorithmic point of view
Guillaume Hanrot,François Morain +1 more
TL;DR: This paper reduces the problem to that of cyclic extensions of prime degree and work out the radicals, using the work of Girstmair, and applies the general framework to the construction of Hilbert Class fields of imaginary quadratic fields.
Journal ArticleDOI
Reductions and simplifications of orbital sums in a Hamiltonian repeller
TL;DR: In this article, it was shown that in the presence of a complete Smale horseshoe, the sum of the coordinates of orbital points for low periodic orbits of the Hamiltonian Henon map reduces to simple rational numbers every so often.
Posted Content
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.
References
More filters
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.
Related Papers (5)
A method for obtaining digital signatures and public-key cryptosystems
Graduate Texts in Mathematics
Rajendra Bhatia,Glen Bredon,Wolfgang Walter,Joseph J. Rotman,M. Ram Murty,Jane Gilman,Peter Walters,Martin Golubitsky,Ioannis Karatzas,Henri Cohen,Raoul Bott,Gaisi Takeuti,Béla Bollobás,John M. Lee,Jiří Matoušek,Saunders Mac Lane,John L. Kelley,B. A. Dubrovin,Tom M. Apostol,John Stillwell,William Arveson +20 more