Algorithmic Number Theory, Vol. 1: Efficient Algorithms (Foundations of Computing) by Eric Bach (PDF)

Description

Algorithmic Number Theory provides a thorough introduction to the design and analysis of algorithms for problems from the theory of numbers. Although not an elementary textbook, it includes over 300 exercises with suggested solutions. Every theorem not proved in the text or left as an exercise has a reference in the notes section that appears at the end of each chapter. The bibliography contains over 1,750 citations to the literature. Finally, it successfully blends computational theory with practice by covering some of the practical aspects of algorithm implementations.The subject of algorithmic number theory represents the marriage of number theory with the theory of computational complexity. It may be briefly defined as finding integer solutions to equations, or proving their non-existence, making efficient use of resources such as time and space. Implicit in this definition is the question of how to efficiently represent the objects in question on a computer. The problems of algorithmic number theory are important both for their intrinsic mathematical interest and their application to random number generation, codes for reliable and secure information transmission, computer algebra, and other areas.Publisher’s Note: Volume 2 was not written. Volume 1 is, therefore, a stand-alone publication.

User’s Reviews

Editorial Reviews: Review “[Algorithmic Number Theory] is an enormous achievement andan extremely valuable reference.” Donald E. Knuth, Emeritus, Stanford University About the Author Eric Bach is Professor, Computer Sciences Department, University of Wisconsin. Jeffrey Shallit is Associate Professor, Department of Computer Science, University of Waterloo.

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

⭐This is a wonderful summary of the standard algorithms used in computational number theory. It’s writing is clear, the proofs are lovely, and the selection of topics is nice.

⭐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.Related algorithms such as the lattice reduction algorithm of Lenstra, Lenstra, and Lovasz, and elliptic curve point counting over finite fields are 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 RH and ERH(Riemann Hypothesis and Extended 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.

⭐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.

⭐It is the treatment of the Riemann hypothesis that really bothers me:mostlythe ERH ( extended Riemann Hypothesis) which is introduced with a badly or incompletely defined phi(n) function).With chapter notes and problem solutions there is less to this book than the size ( page number suggests) so this volume is more of notice by what is left out or left to volume two.The algorithms are all in pseudo-code and there is no bridge to actual computer programs that you might expect from where Jeffrey Shallit teaches( Maple country).I was disappointed with this book and I dislike the abstract mathematics style in a book that is supposed to be about computer algorithms.You are left with a definite “snob without a cause” feeling.

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