Table of Contents:

Preface
Preliminaries

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
Bibliography
Index of notation
Index.

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.