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 GF₄

The Galois Fields GF₈ and GF₁₆

The Galois Field GF_pⁿ

The Multiplicative Group of GF_pⁿ

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