A Course in Computational Algebraic Number Theory (Graduate Texts in Mathematics, 138) by Henri Cohen | (PDF) Free Download

Description

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.

User’s Reviews

Editorial Reviews: Review From the reviews:H. CohenA 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)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A good reference if you are interested in the relevant mathematics, and Cohen is without a doubt an expert in this field. However it was written a long time ago so is basically documenting state of the art circa 1985. Unfortunately, I’m not aware of a more recent work that covers the same ground.

⭐Slightly dated now, but excellent working reference for those who need under the covers code level view of these algorithms. Does not have most recent results from encryption algs, but does have nice discussion of the classical algs and relative performance.

⭐An excellent text for someone interested in implementing encryption protocols, but need to learn the nuts-and-bolts of how key computational ingredients are designed.

⭐This book is an excellent compilation of both the theory and pseudo-code for number theoretic algorithms. The author also takes the time to prove some of the major results as background to the algorithms, in addition to sets of exercises at the end of the book. The book is too large to do a chapter by chapter review, so instead I will list the algorithms in the book that I thought were particularly useful: 1. Most of the algorithms on elliptic curves. The author reminds the reader that number-theoretical experiments resulted in the famous Swinnerton-Dyer Conjecture and the Birch Conjecture. (a) the reduction algorithm, which for a given point in the upper half plane, gives the unique point in the half plane equivalent to this point under the action of the special linear group along with the matrix that maps these two points to each other. (b) The computation of the coefficient g2 and g3 of the Weierstrass equation of an elliptic curve. (c) The computation of the Weierstrass function and its derivative. (d) Determination of the periods of an elliptic curve over the real numbers. (e) The determination of the elliptic logarithm. (f) The reduction of a general cubic (f) The Shanks-Mestre algorithm for computing the order of an elliptic curve over a finite field F(p), where p is prime and greater than 457. (g) The reduction of an elliptic curve modulo p for p > 3. (h) The reduction of an elliptic curve modulo 2 or 3. (i) Reduction of an elliptic curve over the rational numbers. (j) Determination of the rational torsion points of an elliptic curve. (k) Computation of the Hilbert class polynomials and thus a determination of the j-function of an elliptic curve. 2. A few of the algorithms on factoring. (a) The Pollard algorithm for finding non-trivial factors of composites. (The author does not give the improved algorithm due to P. Montgomery, but does give references) (b) Shanks Square Form Factorization algorithm for finding a non-trivial factor of an odd integer. (c) Lenstra’s Elliptic Curve test for compositeness. 3. Primality tests (a) The Jacobi Sum Primality Test for a positive integer. (b) Goldwasser-Killian elliptic curve test for a positive integer not equal to 1 and coprime to 6. The author gives an overview of the computer packages used for number theory, including Pari, which was written by him and his collaborators. I have not used this package, but instead use Lydia and Mathematica for most of the number theoretic computations I need to do.

⭐I bought this book for the math course I had taken having the same title. This is an excellent book, but only if you are really interested in its content. It’s not a casual read, since it’s graduate text. Also, a background in number theory will be of great help – being a CS major, I had a little tough time in the beginning, but things turned out just fine. As for content, it has excellent coverage of the subject, and I would highly recommend this as a reference in this subject. Remember, though, that this book deals COMPUTATIONAL aspects, for only number theory, look for other books like Ireland-Rosen.

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