Applied Algebra Codes, Ciphers and Discrete Algorithms, Second Edition

Using mathematical tools from number theory and finite fields, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition presents practical methods for solving problems in data security and data integrity. It is designed for an applied algebra course for students who have had prior classes in abstract or linear algebra. While the content has been reworked and improved, this edition continues to cover many algorithms that arise in cryptography and error-control codes. New to the Second EditionDownloadable resources containing an interactive version of the book that is powered by Scientific Notebook®, a mathematical word processor and easy-to-use computer algebra systemNew appendix that reviews prerequisite topics in algebra and number theoryDouble the number of exercisesInstead of a general study on finite groups, the book considers finite groups of permutations and develops just enough of the theory of finite fields to facilitate construction of the fields used for error-control codes and the Advanced Encryption Standard. It also deals with integers and polynomials. Explaining the mathematics as needed, this text thoroughly explores how mathematical techniques can be used to solve practical problems. About the AuthorsDarel W. Hardy is Professor Emeritus in the Department of Mathematics at Colorado State University. His research interests include applied algebra and semigroups.Fred Richman is a professor in the Department of Mathematical Sciences at Florida Atlantic University. His research interests include Abelian group theory and constructive mathematics.Carol L. Walker is Associate Dean Emeritus in the Department of Mathematical Sciences at New Mexico State University. Her research interests include Abelian group theory, applications of homological algebra and category theory, and the mathematics of fuzzy sets and fuzzy logic.

Preface Integers and Computer Algebra Integers Computer Algebra vs. Numerical Analysis Sums and Products Mathematical Induction Codes Binary and Hexadecimal Codes ASCII Code Morse Code Braille 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 Ciphers Cryptography Cryptanalysis 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 Groups Rings and Polynomials Fields Linear Algebra and Matrices Solutions to Odd Problems Bibliography Notation Algorithms Figures Tables Index