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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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Journal ArticleDOI
Solving quadratic equations using reduced unimodular quadratic forms
TL;DR: A generalized LLL algorithm to reduce the quadratic form of an n x n symmetric matrix with integral entries and with det Q ¬= 0 is described, which is proved to run in polynomial time.
Book ChapterDOI
Cryptocomputing with rationals
TL;DR: A method to compute with encrypted rational numbers using the Paillier cryptosystem which offers the largest bandwidth among all homomorphic schemes and uses two-dimensional lattices to recover the numerator and denominator of the rationals.
Proceedings ArticleDOI
Quantum algorithm for a generalized hidden shift problem
Andrew M. Childs,Wim van Dam +1 more
TL;DR: For any fixed positive ε, an efficient quantum algorithm is given for the generalized hidden shift problem, based on the "pretty good measurement" and uses H. Lenstra's (classical) algorithm for integer programming as a subroutine.
Posted Content
Algorithmic enumeration of ideal classes for quaternion orders
Markus Kirschmer,John Voight +1 more
TL;DR: In this paper, the authors provide algorithms to count and enumerate representatives of the ideal classes of an Eichler order in a quaternion algebra defined over a number field.
Book ChapterDOI
Short, Invertible Elements in Partially Splitting Cyclotomic Rings and Applications to Lattice-Based Zero-Knowledge Proofs
TL;DR: When constructing practical zero-knowledge proofs based on the hardness of the Ring-LWE or theRing-SIS problems over polynomial rings, it is often necessary that the challenges come from a set that satisfies three properties: the set should be large, the elements in it should have small norms, and all the non-zero elements in the difference set \(\mathcal {C}-\mathcal{C}\) should be invertible.
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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