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 Zp[Xent 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 Zp[Xions) 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 Zp[Xstel 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.