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Introduction to Cryptography with Mathematical Foundations and Computer Implementations

Table of Contents

An Overview of the Subject
Basic Concepts
Functions
One-to-One and Onto Functions, Bijections
Inverse Functions
Substitution Ciphers
Attacks on Cryptosystems
The Vigenère Cipher
The Playfair Cipher
The One-Time Pad, Perfect Secrecy

Divisibility and Modular Arithmetic
Divisibility
Primes
Greatest Common Divisors and Relatively Prime Integers
The Division Algorithm
The Euclidean Algorithm
Modular Arithmetic and Congruencies
Modular Integer Systems
Modular Inverses
Extended Euclidean Algorithm
Solving Linear Congruencies
The Chinese Remainder Theorem

The Evolution of Codemaking until the Computer Era
Ancient Codes
Formal Definition of a Cryptosystem
Affine Ciphers
Steganography
Nulls
Homophones
Composition of Functions
Tabular Form Notation for Permutations
The Enigma Machines
Cycles (Cyclic Permutations)
Dissection of the Enigma Machine into Permutations
Special Properties of All Enigma Machines

Matrices and the Hill Cryptosystem
The Anatomy of a Matrix
Matrix Addition, Subtraction, and Scalar Multiplication
Matrix Multiplication
Preview of the Fact That Matrix Multiplication Is Associative
Matrix Arithmetic
Definition of an Invertible (Square) Matrix
The Determinant of a Square Matrix
Inverses of 2×2 Matrices
The Transpose of a Matrix
Modular Integer Matrices
The Classical Adjoint (for Matrix Inversions)
The Hill Cryptosystem

The Evolution of Codebreaking until the Computer Era
Frequency Analysis Attacks
The Demise of the Vigenère Cipher
The Index of Coincidence
Expected Values of the Index of Coincidence
How Enigmas Were Attacked
Invariance of Cycle Decomposition Form

Representation and Arithmetic of Integers in Different Bases
Representation of Integers in Different Bases
Hex(adecimal) and Binary Expansions
Arithmetic with Large Integers
Fast Modular Exponentiation

Block Cryptosystems and the Data Encryption Standard (DES)
The Evolution of Computers into Cryptosystems
DES Is Adopted to Fulfill an Important Need
The XOR Operation
Feistel Cryptosystems
A Scaled-Down Version of DES
DES
The Fall of DES
Triple DES
Modes of Operation for Block Cryptosystems

Some Number Theory and Algorithms
The Prime Number Theorem
Fermat’s Little Theorem
The Euler Phi Function
Euler’s Theorem
Modular Orders of Invertible Modular Integers
Primitive Roots
Order of Powers Formula
Prime Number Generation
Fermat’s Primality Test
Carmichael Numbers
The Miller–Rabin Test
The Miller–Rabin Test with a Factoring Enhancement
The Pollard p – 1 Factoring Algorithm

Public Key Cryptography
An Informal Analogy for a Public Key Cryptosystem
The Quest for Secure Electronic Key Exchange
One-Way Functions
Review of the Discrete Logarithm Problem
The Diffie–Hellman Key Exchange
The Quest for a Complete Public Key Cryptosystem
The RSA Cryptosystem
Digital Signatures and Authentication
The El Gamal Cryptosystem
Digital Signatures with El Gamal
Knapsack Problems
The Merkle–Hellman Knapsack Cryptosystem
Government Controls on Cryptography
A Security Guarantee for RSA

Finite Fields in General and GF(2⁸) in Particular
Binary Operations
Rings
Fields
Z_p[X] = the Polynomials with Coefficients in Z_p
Addition and Multiplication of Polynomials in Z_p[X]
Vector Representation of Polynomials
Z_p[X] Is a Ring
Divisibility in Z_p[X]
The Division Algorithm for Z_p[X]
Congruencies in Z_p[X] Modulo a Fixed Polynomial
Building Finite Fields from Z_p[X]
The Fields GF(2⁴) and GF(2⁸)
The Euclidean Algorithm for Polynomials

The Advanced Encryption Standard (AES) Protocol
An Open Call for a Replacement to DES
Nibbles
A Scaled-Down Version of AES
Decryption in the Scaled-Down Version of AES
AES
Byte Representation and Arithmetic
The AES Encryption Algorithm
The AES Decryption Algorithm
Security of the AES

Elliptic Curve Cryptography
Elliptic Curves over the Real Numbers
The Addition Operation for Elliptic Curves
Groups
Elliptic Curves over Z_p
The Variety of Sizes of Modular Elliptic Curves
The Addition Operation for Elliptic Curves over Z_p
The Discrete Logarithm Problem on Modular Elliptic Curves
An Elliptic Curve Version of the Diffie–Hellman Key Exchange
Fast Integer Multiplication of Points on Modular Elliptic Curves
Representing Plaintexts on Modular Elliptic Curves
An Elliptic Curve Version of the El Gamal Cryptosystem
A Factoring Algorithm Based on Elliptic Curves

Appendix A: Sets and Basic Counting Principles
Appendix B: Randomness and Probability
Appendix C: Solutions to All Exercises for the Reader
Appendix D: Answers and Brief Solutions to Selected Odd-Numbered Exercises
Appendix E: Suggestions for Further Reading

References

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