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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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References
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Journal ArticleDOI
Solving homogeneous linear equations over GF (2) via block Wiedemann algorithm
TL;DR: A method of solving large sparse systems of homogeneous linear equations over G F ( 2 ) , the field with two elements, is proposed and an algorithm due to Wiedemann is modified, which is competitive with structured Gaussian elimination in terms of time and has much lower space requirements.
Journal ArticleDOI
Explicit bounds for primality testing and related problems
TL;DR: In this article, it was shown that if the Extended Riemann Hypothesis holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log 2m.
Book ChapterDOI
Factoring integers with the number field sieve
TL;DR: In 1990, the 9 Fermat number was factored into primes by means of a new algorithm, the number field sieve, which was proposed by John Pollard as mentioned in this paper.
Book ChapterDOI
Solving Large Sparse Linear Systems over Finite Fields
TL;DR: It is shown that very large sparse systems can be solved efficiently by using combinations of structured Gaussian elimination and the conjugate gradient, Lanczos, and Wiedemann methods.
Journal ArticleDOI
An improved Monte Carlo factorization algorithm
TL;DR: A cycle-finding algorithm is described which is about 36 percent faster than Floyd's (on the average), and applied to give a Monte Carlo factorization algorithm which is similar to Pollard's but about 24 percent faster.
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