**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 *GF*_{p }

The Ring *GF*_{p}[*x*] of Polynomials

The Galois Field *GF*_{4 }

The Galois Fields *GF*_{8} and *GF*_{16 }

The Galois Field *GF*_{p}^{n }

The Multiplicative Group of *GF*_{p}^{n}

Random Number Generators

**Error-Correcting Codes**

BCH Codes

A BCH Decoder

Reed–Solomon Codes

**Advanced Encryption Standard**

Data Encryption Standard

The Galois Field *GF*_{256 }

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**