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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
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Journal ArticleDOI
A fast algorithm to compute cubic fields
Karim Belabas,Karim Belabas +1 more
TL;DR: A very fast algorithm to build up tables of cubic fields with discriminant up to 10 11 and complex cubic fields down to -10 11 has been computed.
Journal ArticleDOI
General Secret Sharing Based on the Chinese Remainder Theorem with Applications in E-Voting
TL;DR: In this paper, Asmuth et al. showed that the Chinese remainder theorem can be used for realizing more general access structures, such as the compartmented or the weighted threshold ones.
Book ChapterDOI
Zero-Value Point Attacks on Elliptic Curve Cryptosystem
Toru Akishita,Tsuyoshi Takagi +1 more
TL;DR: In this article, the authors proposed the zero-value point attack as an extension of Goubin's attack and demonstrated the standard curves that have these points and showed that these points are not applicable to ECDSA signature generation.
MonographDOI
Algebraic Shift Register Sequences
Mark Goresky,Andrew Klapper +1 more
TL;DR: In this paper, the authors present algebraically defined sequences and their correlations with linear feedback shift registers and linear recurrences, and Maximal Period Function Field Sequence (M-FCSR) sequences.
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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