A Computational Introduction to Number Theory and Algebra

Victor Shoup

518 pages, parution le 11/05/2005

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Résumé

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

Sommaire

Preliminaries
Basic properties of the integers
Congruences
Computing with large integers
Euclid's algorithm
The distribution of primes
Finite and discrete probability distributions
Probabilistic algorithms
Abelian groups
Rings
Probabilistic primality testing
Finding generators and discrete logarithms in Z^*_p
Quadratic residues and quadratic reciprocity
Computational problems related to quadratic residues
Modules and vector spaces
Matrices
Subexponential-time discrete logarithms and factoring
More rings
Polynomial arithmetic and applications
Linearly generated sequences and applications
Finite fields
Algorithms for finite fields
Deterministic primality testing
Appendix: some useful facts
Bibliography
Index of notation
Index

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	PAPIER
Éditeur(s)	Cambridge University Press
Auteur(s)	Victor Shoup
Parution	11/05/2005
Nb. de pages	518
Format	18 x 25,5
Couverture	Relié
Poids	1208g
Intérieur	Noir et Blanc
EAN13	9780521851541
ISBN13	978-0-521-85154-1