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.

0. Preliminaries 1. Basic properties of the integers 2. Congruences 3. Computing with large integers 4. Euclid's algorithm 5. The distribution of primes 6. Finite and discrete probability distributions 7. Probabilistic algorithms 8. Abelian groups 9. Rings 10. Probabilistic primality testing 11. Finding generators and discrete logarithms in Zp* 12. Quadratic residues and quadratic reciprocity 13. Computational problems related to quadratic residues 14. Modules and vector spaces 15. Matrices 16. Subexponential-time discrete logarithms and factoring 17. More rings 18. Polynomial arithmetic and applications 19. Linearly generated sequences and applications 20. Finite fields 21. Algorithms for finite fields 22. Deterministic primality testing Appendix some useful facts Bibliography Index of notation Index.

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