A computational introduction to number theory and algebra
- 2nd ed.
- Cambridge Cambridge University Press c2009
- xvii, 580p.; 25cm.
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.
9780521516440
Computer science--Mathematical principles Computer algebra Computational geometry Mathematics--Algorithmics--Complexity Information theory and coding