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Modern electronic commerce requires the secure, error-free collection and transmission of data. Basic number theory provides the tools needed to perform these operations and this book is an excellent introduction to the theory behind it all. The teaching of mathematics generally suffers from a chronic affliction. Where the examples are easily understandable they are not practical or useful and where they are practical and useful they are not easily understandable. That is not the case with the demonstrations in this book. We all have encountered the ISBN numbers of books, the Universal Product Code (UPC) on products and most of us own charge cards. The simple, yet effective ways in which number theory is used to verify the correctness of these numbers is one of the most demonstrable uses of mathematics that exists in the world today. Furthermore, the level of mathematics is well within the reach of an advanced high school or first year college student. The most advanced topic in the book is the most basic of group theoretic concepts. The book is structured so that it could be used as a textbook, and there are three classes of problems at the end of each chapter. A first set of basic exercises that reinforce the discussion of the chapter, paper assignments, where topics for potential short papers are given and finally activities that can be done in a group. The exercises are well-done and test the reader’s understanding of the material of the chapter. This is a superb book on many levels. The best aspect is that it shows how mathematics is used in practical applications, where the background is simple enough for almost anyone to follow. Truly one of the best demonstrations of the utility of mathematics that exists today. Published in Mathematics and Computer Education, reprinted with permission.
Introduction
To the Student
To the Professor
Identification Numbers and Check Digit Schemes
Developing Identification Number
Types of Identification Numbers
Transmission Errors
Check Digits
Number Theory, Check Digit Schemes, and Cryptography
Preliminaries
Integer Division
Modulo Arithmetic
US Postal Money Orders
Airline Ticket Identification Numbers
The Universal Product Code Check Digit Scheme
The ISBN Check Digit Scheme
Cryptography and the RSA Public-Key System
Functions, Permutations, and Their Applications
Sets
Creating Identification Numbers
Functions
Permutations
The IBM Scheme
Graphs of Functions
Symmetry and Rigid Motions
Symmetry
Symmetry and Rigid Motions
Group Theory and the Verhoeff Check Digit Scheme
Fundamental Concepts
Cayley Tables
Powers and Orders of Group Elements
The Verhoeff Check Digit Scheme
Bibliography
Index