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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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Proceedings ArticleDOI
Almost all primes can be quickly certified
Shafi Goldwasser,Joe Kilian +1 more
TL;DR: A new probabilistie primality test is presented, different from the tests of Miller, Solovay-Strassen, and Rabin in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.
Journal ArticleDOI
A method of factoring and the factorization of
TL;DR: The continued fraction method for factoring integers, introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation in this paper, and the power of the method is demonstrated by the factorization of the seventh Fermat number F7 and other large numbers of interest.
Proceedings ArticleDOI
Factoring polynomials over large finite fields
TL;DR: Some of the known algorithms for factoring polynomials over finite fields are reviewed and a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p m) is presented.
Proceedings ArticleDOI
Algebraic factoring and rational function integration
TL;DR: A new, simple, and efficient algorithm for factoring polynomials in several variables over an algebraic number field is presented and a constructive procedure is given for finding the least degree extension field in which the integral can be expressed.
Journal ArticleDOI
A rigorous subexponential algorithm for computation of class groups
TL;DR: In this paper, McCurley gave a rigorous proof that there exists a Las Ve- gas algorithm that will compute the structure of C(-d) with an expected running time of L(d) vf2+?(1) bit operations.
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