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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
Computing canonical heights with little (or no) factorization
TL;DR: A method for computing the non-archimedean contribution to h(P) which is quite practical and requires little or no factorization.
Proceedings ArticleDOI
Improvement over Public Key Cryptographic Algorithm
Deepak Garg,Seema Verma +1 more
TL;DR: The proposed approach to improve decryption/signature generation speed is the improvement by the combination of MultiPower RSA and Rebalanced RSA, and theoretically, the proposed scheme is about 14 times faster than that given by RSA with CRT and about 56 times fasterthan the standard RSA.
Book ChapterDOI
An improvement of key generation algorithm for Gentry's homomorphic encryption scheme
TL;DR: A key generation algorithm is proposed for Gentry's homomorphic encryption scheme that controls the bound of the circuit depth by using the relation between the circuit Depth and the eigenvalues of a basis of a lattice.
Journal ArticleDOI
Linear relations of zeroes of the zeta-function
TL;DR: This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function and presents an alternative disproof to Mertens’ conjecture by showing that limsupx!1 M(x)x 1=2 1:6383, and liminfx!2 M( x)X 1=1 1: 6383.
Optimizing curve-based cryptography
TL;DR: The final author version and the galley proof are versions of the publication after peer review and the final published version features the final layout of the paper including the volume, issue and page numbers.
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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