A Course in Number Theory and Cryptography

by Koblitz, Neal

• ISBN13: 9780387942933
• ISBN10: 0387942939
• Edition: 2nd
• Format: Hardcover
• Copyright: 1994-08-01
• Publisher: Springer Verlag

Summary

The purpose of this book is to introduce the reader to arithmetic topics, both ancient and modern, that have been at the center of interest in applications of number theory, particularly in cryptography. No background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasizing estimates of the efficiency of the techniques that arise from the theory. A special feature is the inclusion of recent application of the theory of elliptic curves. Extensive exercises and careful answers have been included in all of the chapters. Because number theory and cryptography are fast-moving fields, this new edition contains substantial revisions and updated references.

Table of Contents

Foreword	v
Preface to the Second Edition	vii
Some Topics in Elementary Number Theory	1	(30)
Time estimates for doing arithmetic	1	(11)
Divisibility and the Euclidean algorithm	12	(7)
Congruences	19	(8)
Some applications to factoring	27	(4)
Finite Fields and Quadratic Residues	31	(23)
Finite Fields	33	(9)
Quadratic residues and reciprocity	42	(12)
Cryptography	54	(29)
Some simple cryptosystems	54	(11)
Enciphering matrices	65	(18)
Public Key	83	(42)
The idea of public key cryptography	83	(9)
RSA	92	(5)
Discrete log	97	(14)
Knapsack	111	(6)
Zero-knowledge protocols and oblivious transfer	117	(8)
Primality and Factoring	125	(42)
Pseudoprimes	126	(12)
The rho method	138	(5)
Fermat factorization and factor bases	143	(11)
The continued fraction method	154	(6)
The quadratic sieve method	160	(7)
Elliptic Curves	167	(33)
Basic facts	167	(10)
Elliptic curve cryptosystems	177	(10)
Elliptic curve primality test	187	(4)
Elliptic curve factorization	191	(9)
Answers to Exercises	200	(31)
Index	231