A good introduction to classical and modern number theory and its applications in computer science
Self-contained source on number theory for computing professionals
Useful for self-study or as class text and basic reference
Only prerequisite is high-school math
First edition released in 2000
In the second edition additional proofs of many theorems are provided, corrections and additions were made
This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, including computer systems design, cryptography and network security. In this 2nd edition proofs of many theorems have been provided, further additions and corrections were made.
Foreword by Martin E. Hellman.- Preface to the Second Edition.- Preface to the First Edition.- 1. Elementary Number Theory.- 2. Computational/Algorithmic Number Theory.- 3. Applied Number Theory.- Bibliography.- Index.
There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing and information theory. Number theory has important applications in computer organization and security, coding and cryptography, random number generation, hash functions, and graphics. Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. This book takes the reader from elementary number theory, via algorithmic number theory, to applied number theory in computer science. It introduces basic concepts, results, and methods, and discusses their applications in the design of hardware and software, cryptography, and security. It is aimed at undergraduates in computing and information technology, including electrical and electronic engineering, but will also be valuable to mathematics students interested in applications. It presupposes only high-shool math.
In this 2nd edition proofs of many theorems are added and some corrections are made.
From the reviews of the second edition:
"This book gives a profound and detailed insight at an undergraduate level in abstract and computational number theory as well as in applications in computing and cryptography. ... The author has done a lot of work in providing a plenty of examples, in adding many historical comments including sketchy biographies ... and in presenting the whole topic in a very accessible style. So the book can be recommended warmly for the laymen as well as for the mathematician without experience in applied number theory." (G. Kowol, Monatshefte für Mathematik, Vol. 140 (4), 2003)
Modern cryptography depends heavily on number theory, with primality test ing, factoring, discrete logarithms (indices), and elliptic curves being perhaps the most prominent subject areas. Since my own graduate study had empha sized probability theory, statistics, and real analysis, when I started work ing in cryptography around 1970, I found myself swimming in an unknown, murky sea. I thus know from personal experience how inaccessible number theory can be to the uninitiated. Thank you for your efforts to case the transition for a new generation of cryptographers. Thank you also for helping Ralph Merkle receive the credit he deserves. Diffie, Rivest, Shamir, Adleman and I had the good luck to get expedited review of our papers, so that they appeared before Merkle's seminal contribu tion. Your noting his early submission date and referring to what has come to be called "Diffie-Hellman key exchange" as it should, "Diffie-Hellman-Merkle key exchange", is greatly appreciated. It has been gratifying to see how cryptography and number theory have helped each other over the last twenty-five years. :'-Jumber theory has been the source of numerous clever ideas for implementing cryptographic systems and protocols while cryptography has been helpful in getting funding for this area which has sometimes been called "the queen of mathematics" because of its seeming lack of real world applications. Little did they know! Stanford, 30 July 2001 Martin E. Hellman Preface to the Second Edition Number theory is an experimental science.
From the contents:
- Foreword by Martin E. Hellman
- Preface1. Elementary Number Theory
2. Computational/Algorithmic Number Theory
3. Applied Number Theory
- Bibliography
- Index
From the reviews of the second edition:
"This book gives a profound and detailed insight at an undergraduate level in abstract and computational number theory as well as in applications in computing and cryptography. ... The author has done a lot of work in providing a plenty of examples, in adding many historical comments including sketchy biographies ... and in presenting the whole topic in a very accessible style. So the book can be recommended warmly for the laymen as well as for the mathematician without experience in applied number theory." (G. Kowol, Monatshefte für Mathematik, Vol. 140 (4), 2003)
Inhaltsverzeichnis
Foreword by Martin E. Hellman.- Preface to the Second Edition.- Preface to the First Edition.- 1. Elementary Number Theory.- 2. Computational/Algorithmic Number Theory.- 3. Applied Number Theory.- Bibliography.- Index.
Klappentext
Modern cryptography depends heavily on number theory, with primality test ing, factoring, discrete logarithms (indices), and elliptic curves being perhaps the most prominent subject areas. Since my own graduate study had empha sized probability theory, statistics, and real analysis, when I started work ing in cryptography around 1970, I found myself swimming in an unknown, murky sea. I thus know from personal experience how inaccessible number theory can be to the uninitiated. Thank you for your efforts to case the transition for a new generation of cryptographers. Thank you also for helping Ralph Merkle receive the credit he deserves. Diffie, Rivest, Shamir, Adleman and I had the good luck to get expedited review of our papers, so that they appeared before Merkle's seminal contribu tion. Your noting his early submission date and referring to what has come to be called "Diffie-Hellman key exchange" as it should, "Diffie-Hellman-Merkle key exchange", is greatly appreciated. It has been gratifying to see how cryptography and number theory have helped each other over the last twenty-five years. :'-Jumber theory has been the source of numerous clever ideas for implementing cryptographic systems and protocols while cryptography has been helpful in getting funding for this area which has sometimes been called "the queen of mathematics" because of its seeming lack of real world applications. Little did they know! Stanford, 30 July 2001 Martin E. Hellman Preface to the Second Edition Number theory is an experimental science.
This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, including computer systems design, cryptography and network security. In this 2nd edition proofs of many theorems have been provided, further additions and corrections were made.
