An Extension Of The Large Sieve To Algebraic Number Fields
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The Development of the Number Field Sieve
| Author | : Arjen K. Lenstra |
| Publisher | : Springer |
| Total Pages | : 138 |
| Release | : 2006-11-15 |
| Genre | : Mathematics |
| ISBN | : 3540478922 |
The number field sieve is an algorithm for finding the prime factors of large integers. It depends on algebraic number theory. Proposed by John Pollard in 1988, the method was used in 1990 to factor the ninth Fermat number, a 155-digit integer. The algorithm is most suited to numbers of a special form, but there is a promising variant that applies in general. This volume contains six research papers that describe the operation of the number field sieve, from both theoretical and practical perspectives. Pollard's original manuscript is included. In addition, there is an annotated bibliography of directly related literature.
Elementary and Analytic Theory of Algebraic Numbers
| Author | : Wladyslaw Narkiewicz |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 712 |
| Release | : 2013-06-29 |
| Genre | : Mathematics |
| ISBN | : 3662070014 |
This book details the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. Coverage includes: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. The book also features exercises and a list of open problems.
The Development of the Number Field Sieve
| Author | : Arjen K. Lenstra |
| Publisher | : Springer |
| Total Pages | : 140 |
| Release | : 1993-08-30 |
| Genre | : Mathematics |
| ISBN | : 9783540570134 |
The number field sieve is an algorithm for finding the prime factors of large integers. It depends on algebraic number theory. Proposed by John Pollard in 1988, the method was used in 1990 to factor the ninth Fermat number, a 155-digit integer. The algorithm is most suited to numbers of a special form, but there is a promising variant that applies in general. This volume contains six research papers that describe the operation of the number field sieve, from both theoretical and practical perspectives. Pollard's original manuscript is included. In addition, there is an annotated bibliography of directly related literature.
The Development of the Number Field Sieve
| Author | : Arjen K. Lenstra |
| Publisher | : |
| Total Pages | : 148 |
| Release | : 2014-01-15 |
| Genre | : |
| ISBN | : 9783662167045 |
Algebraic Number Fields
| Author | : |
| Publisher | : Academic Press |
| Total Pages | : 233 |
| Release | : 1973-08-15 |
| Genre | : Mathematics |
| ISBN | : 0080873707 |
Algebraic Number Fields
Pollard's Number Field Sieve
| Author | : Alexander Shaumyan |
| Publisher | : |
| Total Pages | : 252 |
| Release | : 1998 |
| Genre | : Algebraic number theory |
| ISBN | : |
The Number Field Sieve has revolutionized the field of computational number theory and enabled researchers to factor very large integers--a task which seemed insurmountable with the best computers and the best factorization techniques that were available a few years earlier. We'll illustrate how the Number Field Sieve works, using various examples of factorizations in different number fields. We'll also see how the method can be extended to factor any integer m, not just a special m which can be written in the form m = r"--S for which the original algorithm was developed. We'll look at the modified version of the Number Field Sieve known as the General Number Field Sieve (GNFS) and look at an example of a number factored with GNFS. But in order to understand the process, we need to introduce the concepts of algebraic number fields and algebraic numbers.
A Course in Computational Algebraic Number Theory
| Author | : Henri Cohen |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 556 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 3662029456 |
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.
