A Course in Computational Algebraic Number Theory

1. Front Matter

   Pages I-XXI

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2. Fundamental Number-Theoretic Algorithms

   • Henri Cohen

   Pages 1-44

3. Algorithms for Linear Algebra and Lattices

   • Henri Cohen

   Pages 45-107

4. Algorithms on Polynomials

   • Henri Cohen

   Pages 108-150

5. Algorithms for Algebraic Number Theory I

   • Henri Cohen

   Pages 151-217

6. Algorithms for Quadratic Fields

   • Henri Cohen

   Pages 218-296

7. Algorithms for Algebraic Number Theory II

   • Henri Cohen

   Pages 297-359

8. Introduction to Elliptic Curves

   • Henri Cohen

   Pages 360-411

9. Factoring in the Dark Ages

   • Henri Cohen

   Pages 412-436

10. Modern Primality Tests

    • Henri Cohen

    Pages 437-468

11. Modern Factoring Methods

    • Henri Cohen

    Pages 469-497

12. Back Matter

    Pages 498-536

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With the advent of powerful computing tools and numerous advances in math­ ematics, computer science and cryptography, algorithmic number theory has become an important subject in its own right. Both external and internal pressures gave a powerful impetus to the development of more powerful al­ gorithms. These in turn led to a large number of spectacular breakthroughs. To mention but a few, the LLL algorithm which has a wide range of appli­ cations, including real world applications to integer programming, primality testing and factoring algorithms, sub-exponential class group and regulator algorithms, etc ... Several books exist which treat parts of this subject. (It is essentially impossible for an author to keep up with the rapid pace of progress in all areas of this subject.) Each book emphasizes a different area, corresponding to the author's tastes and interests. The most famous, but unfortunately the oldest, is Knuth's Art of Computer Programming, especially Chapter 4. The present book has two goals. First, to give a reasonably comprehensive introductory course in computational number theory. In particular, although we study some subjects in great detail, others are only mentioned, but with suitable pointers to the literature. Hence, we hope that this book can serve as a first course on the subject. A natural sequel would be to study more specialized subjects in the existing literature.

• Algorithms
• Computer Algebra
• Factorisation
• Primality
• Symbolic Computation
• algebra
• algorithm
• computer
• number theory
• algorithm analysis and problem complexity

From the reviews:

H. Cohen

A Course in Computational Algebraic Number Theory

"With numerous advances in mathematics, computer science, and cryptography, algorithmic number theory has become an important subject. Undoubtedly, this book, written by one of the leading authorities in the field, is one of the most beautiful books available on the market."

—ACTA SCIENTIARUM MATHEMATICARUM

“This book is intended to provide material for a three-semester sequence, introductory, graduate course in computational algebraic number theory. … The book is excellent. … The book has 75 sections, making it suitable for a three-semester sequence. There are numerous exercises at all levels … . The bibliography is quite comprehensive and therefore has intrinsic value in its own right. … chapters bring the student to the frontiers of the field, covering elliptic curves, modern primality testing and modern factoring methods.” (Russell Jay Hendel, The Mathematical Association of America, January, 2011)

• U.F.R. de Mathématiques et Informatique, Université Bordeaux I, Talence Cedex, France

  Henri Cohen