A Computational Introduction to Number Theory and Algebra

This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

Table of Contents

1 Basic properties of the integers

2 Congruences

3 Computing with large integers

4 Euclid’s algorithm

5 The distribution of primes

6 Abelian groups

7 Rings

8 Finite and discrete probability distributions

9 Probabilistic algorithms

10 Probabilistic primality testing

11 Finding generators and discrete logarithms in Z_p*

12 Quadratic reciprocity and computing modular square roots

13 Modules and vector spaces

14 Matrices

15 Subexponential-time discrete logarithms and factoring

16 More rings

17 Polynomial arithmetic and applications

18 Linearly generated sequences and applications

19 Finite fields

20 Algorithms for finite fields

21 Deterministic primality testing

Appendix: Some useful facts

Review

“The subject matter is presented in a very thorough way. Detailed and clear proofs are given for (almost) all results. The material is very well organized: definitions, results, and their interrelations fit together perfectly. The book is especially attractive to students with a background or interest in computer science. On the other hand, students who already have a background in abstract algebra can benefit greatly from this book by skipping some parts where algebraic theory is introduced. The suitability of the book for self-study is greatly enhanced by a wealth of exercises and examples that are provided. Last but not least, the book is very well-written, and it is a pleasure to read.

Gregor Kemper, Mathematics of Computation

“It’s a pleasure to find a book that is so masterful and so well written that it has all the hallmarks of a classic. This is such a book.This is a truly magnificent text, deserving of a place on the shelves of any mathematician or computer scientist working in these areas.”

Alasdair McAndrew, Computing Reviews

Book Description

An introductory graduate-level text emphasizing algorithms and applications. This second edition includes over 200 new exercises and examples.

About the Author

Victor Shoup is a Professor in the Department of Computer Science at the Courant Institute of Mathematical Sciences, New York University.