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Rating: 
Summary: Worthless!
Review: As with many math books, the author immediately begins by using some peculiar looking boldfaced symbols in an outlined font (Z, Q, R, C, etc) without ever once explaining what they are supposed to represent. This should be a criminal offense.
Rating: 
Summary: Excellent!
Review: Cohen (the world renowned expert) starts with the most basic of algorithms (i.e. Euclid & Shanks). He moves seamlessly into Linear Algebra & Polynomials (bedrocks of most CAS). Although meant to be concise, he proves, or sketches a proof of the important results. Finally, the meat of the book, C.A.N.T. One important problem is finding the "class number" (has to do with unique factorization, which we are all accustomed to in Z). A detailed description of the continued fraction algorithm (for finding the fundamental unit), and others made it very enlightening. He then deals with primality testing and factoring, two very important problems, the latter because of RSA. First, a description of the algorithm, then the theory behind it. He covered everything, from Trial Division (Dark Ages) to Pollard Rho to NFS (cutting-edge). Also included are some useful tables.Of course, CAS information from 1993, won't be that helpful (look in his newest, Advanced Topics in C.A.N.T.). Excellent. Also try Knuth's "Semi-numerical Algorithms" for a more computer oriented approach.
Rating:
Summary: Definitely belongs on the shelf of all number theory lovers
Review: 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.
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