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.
Contains over 450 exercises, which present new applications to number theory and algebra
Minimal mathematics prerequisites
Presents complete and self-contained proofs of Chebyshev’s theorem on the distribution of primes, Bertand’s postulate, Mertens’ theorem and the leftover hash lemma
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. 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;