Algorithmic Number Theory, Vol. 1: Efficient Algorithms (Foundations of Computing)

Rating:
Summary: Excellent Reference
Review: Bach and Shallit have done a wonderful job of preparing a survey of Number Theoretic Algorithms. After covering the basic mathematical material and complexity theory background, the book plunges in to discuss computation in (Z/(n)) and various algorithms in Finite Fields.

The part of the book that I like best are the last two chapters which deal with prime numbers and algorithms for primality testing. The authors have done an exhaustive survey of this area. Proofs of the correctness of the algorithms are wonderfully concise and lucid. The second volume [not published yet] will discuss problems for which efficient algorithms are currently unknown for example factoring, discrete log etc. The authors also promise coverage of the Adleman, Huang proof that Primes \in ZPP.

Exercises have been chosen carefully, and most of the solutions are available as an appendix (for the others references are given). Finally the bibliography is *huge* with close to 2000 citations. Overall an excellent book for reference and for a one stop introduction to the wonderful area of Algorithmic Number Theory.

Rating:
Summary: A work of outstanding mathematical scholarship
Review: This book is a valuable reference -- a real work of mathematical scholarship concerning problems from elementary number theory, such as primality testing, square roots mod p, quadratic residues, polynomial factoring, and generation of random primes -- algorithms for which efficient solutions are known. However, the lattice reduction algorithm of Lenstra, Lenstra, and Lovasz is not covered.

Three outstanding features of this book are:

1) The extensive chapter end notes that provide a comprehensive review of the history and state of the art for each topic addressed in the book. These notes are so detailed that they are like having a mini book within a book. Anyone doing research in the field would do well to own this book for this reason alone.

2) Exhaustive bibliography, all together there are over 1750 bibliographic entries.

3) Applications of the ERH/ GRH (Extended and Generalized Riemann Hypothesis). I know of no other single reference that covers the consequences of these conjectures being true in terms of primality testing, quadratic non residue testing, primitive root finding and so on.

The algorithms are presented in pseudo code and practical implementation remarks are reserved for the notes section of each chapter.

Recommended for upper level undergraduates and all the way on up to faculty.

As a bonus the book is a real pleasure to view due to the excellent job done in the layout and typesetting.

I look forward to volume two which will focus on algorithms for intractable problems for which efficient (polynomial time) algorithms are NOT known such as factoring and the discrete log problem.