Number Theory with Applications in Cryptology

Marc's cyptology page

Instructor

Marc Renault

Description

We will use the problems of making and breaking codes as motivation to learning many of the basics of number theory. We will start by studying the simple shift codes and end the course by looking at public-key cryptology which is used extensively in modern electronic communication. Along the way we'll study classical methods of cryptography, modular arithmetic, the Euclidean algorithm, linear congruences, Fermat's Little Theorem, and Euler's phi function. The properties of prime numbers and divisibility will be central to the course.

Course Outline

• Classical Crpytology
  • Permutation Ciphers
    • The Spartan Scytale
    • The Rail-Fence Cipher
    • The Grille
    • The Transposition Cipher
  • Monoalphabetic Substitution Ciphers
    • The Shift Cipher
    • The Affine Cipher
      • Introduction to modular arithmetic
      • Solving linear congruences
      • Euclidean Algorithm
    • The Keyword Cipher
  • Polyalphabetic Substitution Ciphers
    • Equalizing the Frequencies
      • Frequency analyzer
    • The Vigenere Cipher
    • Jefferson's Wheel
    • The Hill Cipher
    • The Playfair Cipher
    • The Double Playfair Cipher
    • The One-Time Pad
    • Pseudorandom Sequences
    • The Enigma Machine
• Public Key Cryptosystems
  • Diffie-Hellman Key Agreement
    • Some More Modular Arithmetic
    • The Discrete Log Problem
    • The Key Agreement Scheme
  • The RSA Algorithm
    • Fermat's Little Theorem
    • Euler's Phi-function
    • Euler's Theorem
    • The Asymmetric Key Concept and RSA Implementation
  • Other Issues
    • Authenticity
    • Certification Authorities
    • Other Public Key Cryptosystems

Please see this list of Other Resources for some great web sites and books.