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Applied Algebra: Codes, Ciphers, and Discrete Algorithms

D. W. Hardy, F. Richman, and C. L. Walker
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Hardcover with CDROM
Discrete Mathematics and Its Applications
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Gizem Karaali
, on

This book is an introduction to algebra and cryptography. It does not pretend to go deep in either direction, but provides an excellent overview of both. Looking through the table of contents the readers will notice some quite intriguing topics. I have seen other algebra texts which mention Hamming codes or public key cryptography or the discrete Fourier transform, but I have not read about Braille or Morse code in a mathematical context elsewhere. Similarly, perhaps barcodes show up in some texts, but I have not seen such extensive study of them anywhere else. This book is full of interesting applications of elementary abstract algebra and number theory, and these are not included as an afterthought; they make up the core of the text.

The authors introduce many diverse ideas and examples of codes and ciphers throughout; the abstract algebra/elementary number theory material is sprinkled in as needed to make the theory comprehensible. The focus of the book is on examples and hands-on exercises. Each chapter is split into several brief sections which are followed by approximately ten exercises each. The sections are really brief, a section of five pages would be one of the longer sections. The style is relatively informal, and the density of theorems and proofs is rather low. Thus even a five-page section is really easy to read fast and digest if the reader is working along the examples with a pen or, in some cases, with a CAS at hand. The mathematical maturity needed is not high at all and the application-oriented approach could resonate with many different sets of students.

Who could be a good audience for such a book? Of course if your college or university offers a course in applied algebra or elementary cryptography, then this book may be a good choice for a textbook for such a course. (Those intrigued by this possibility may also like to look at Protecting Information: From Classical Error Correction to Quantum Cryptography by Susan Loepp and William Wootters.) However, applied algebra is not a standard part of the undergraduate curriculum in the United States. Therefore readers may like to know how this book may fit into more traditional courses. Instructors teaching abstract algebra or number theory can find many interesting applications and computational exercises in this book that would enrich their courses. More importantly, this book could conceivably be used in a more traditional abstract algebra course for future teachers. All students benefit from seeing applications of abstractions they are studying, but exposing future teachers to this material is likely to have even more impact.

I have found many interesting tidbits of information and lots of exciting things about the book overall. However I have not been able to use the CD-ROM that comes with the book. This CD-ROM includes the full text of the whole book, as well as several interactive features such as exercises, and self-quizzes; the list of such examples is quite impressive and promises to add much to the student experience. Unfortunately I have not been able to view any of this because the CD-ROM employs a word-processor/CAS (Scientific Notebook) specifically designed for PCs (though it is supposed to work on Apple computers that emulate a Virtual PC environment). This makes the CD-ROM useless for many who, like me, are exclusively bound to a Mac platform. The authors are aware of this issue however, and are hoping to provide Mac and Linux versions for the supplementary material in the near future. In the interim, a publisher supported web site could be useful to provide at least some of this supplemental material.

Nonetheless I will finish this review on a high note. Despite the above-mentioned issues of access to the supplementary material, this book is strong enough to stand on its own. While still an undergraduate I worked as a summer intern at a cryptography research lab, but many years have passed since I have really been excited about crypto-stuff. This book brings back the excitement and offers to add it on to a standard algebra course. Well worth checking out!

Gizem Karaali is assistant professor of mathematics at Pomona College. She is a Macoholic and is not apologetic about it (except when talking to Linux users).



Integers and Computer Algebra


Computer Algebra vs. Numerical Analysis

Sums and Products

Mathematical Induction


Binary and Hexadecimal Codes


Morse Code


Two-out-of-Five Code

Hollerith Codes

Euclidean Algorithm

The Mod Function

Greatest Common Divisors

Extended Euclidean Algorithm

The Fundamental Theorem of Arithmetic

Modular Arithmetic




Substitution and Permutation Ciphers

Block Ciphers

The Playfair Cipher

Unbreakable Ciphers

Enigma Machine

Error-Control Codes

Weights and Hamming Distance

Bar Codes Based on Two-out-of-Five Code

Other Commercial Codes

Hamming (7, 4) Code

Chinese Remainder Theorem

Systems of Linear Equations Modulo n

Chinese Remainder Theorem

Extended Precision Arithmetic

Greatest Common Divisor of Polynomials

Hilbert Matrix

Theorems of Fermat and Euler

Wilson’s Theorem

Powers Modulo n

Fermat’s Little Theorem

Rabin’s Probabilistic Primality Test

Exponential Ciphers

Euler’s Theorem

Public Key Ciphers

The Rivest–Shamir–Adleman Cipher System

Electronic Signatures

A System for Exchanging Messages

Knapsack Ciphers

Digital Signature Standard

Finite Fields

The Galois Field GFp

The Ring GFp[x] of Polynomials

The Galois Field GF4

The Galois Fields GF8 and GF16

The Galois Field GFpn

The Multiplicative Group of GFpn

Random Number Generators

Error-Correcting Codes

BCH Codes

A BCH Decoder

Reed–Solomon Codes

Advanced Encryption Standard

Data Encryption Standard

The Galois Field GF256

The Rijndael Block Cipher

Polynomial Algorithms and Fast Fourier Transforms

Lagrange Interpolation Formula

Kronecker’s Algorithm

Neville’s Iterated Interpolation Algorithm

Secure Multiparty Protocols

Discrete Fourier Transforms

Fast Fourier Interpolation

Appendix A: Topics in Algebra and Number Theory

Number Theory


Rings and Polynomials


Linear Algebra and Matrices

Solutions to Odd Problems