1. Elementary Number Theory
   • Fundamental Theorem of Arithmetic
   • lcm, gcd
   • Euclidean Algorithm for gcd
   • Extended Euclidean Algorithm
   • Modular Arithmetic
   • Groups, Abelian Groups, Rings and Fields
   • Modular Inverses
   • Chinese Reminder Theorem
   • Modular Matrix Inversion
2. Classical Cryptography
   • Substitution Cipher
   • Hill Cipher
   • Vignere Ciphere
   • Permutation Cipher
   • Cryptanalysis of Classical Ciphers
3. Theoretical Foundations
   • Perfect Secrecy
   • One-time pad
   • Entropy
   • Confusion and Diffusion
4. Symmetric Cryptography
   • Product Cryptosystems
   • Substitution-Permutation Networks
   • Fiestel Ciphers
   • DES, AES
   • Overview of Linear and Differential Cryptanalysis
   • Stream Ciphers
   • Block Cipher Modes - ECB, CFB, CBC, OFB
   • Double DES and Triple DES
5. Hash Functions
   • Preimage resistance and Collision Resistance
   • Compression Function
   • Merkle-Damgard Construction
   • SHA
   • Hash Functions as Block Ciphers
   • Block Ciphers as Hash Functions
6. Message Authentication Codes
7. Key Distribution and Authentication
   • Basic Key Distribution
   • Diffie-Hellman
   • Blom's Key Predistribution
   • Authentication with a shared secret
   • Kerberos
   • Authentication with a "weak" secret
   • EKE
8. Advanced Topics in Number Theory
   • Euler-Phi Function
   • Fermat's Little Theorem
   • Euler-Fermat Theorem
   • Quadratic Congruences
     • Quadratic congruences modulo a prime.
     • Quadratic congruences modulo composites.
     • Quadratic congruences and Factorization
   • Factorization
     • Fermat's Factorization
     • Pollard's p-1 Factorization
     • Dixon's Random Squares and Quadratic Seives
   • Primality Testing
     • Using Fermat's Little Theorem
     • Miller-Rabin
   • Exponential Congruences
     • Order of elements in a field
     • Generators
     • Discrete logs
     • Algorithms for Discrete Log
       • Pohlig-Hellman
       • Index Calculus Method
9. Cryptosystems based on Infeasibility of Factoring
   • RSA
     • Square and Multiply Algorithm
     • System renewal with compromised private key
     • Small exponent RSA
     • Common modulus RSA
   • Rabin Cryptosystem
10. Cryptosystems based on Infeasibility of the Discrete Log Problem
    • Diffie-Hellman
    • Massey-Omura
    • El Gamal
11. Signature Schemes
    • El Gamal Signature Scheme
    • Schnorr Signature Scheme
    • DSA
12. Brief Overview of Elliptic Curve Cryptosystems

Assignments

• Unzip "ciphertext.zip"
• Choose the ascii file with your LASTNAME.txt
• Break the Vignere Cipher encoded text
• Email me the plaintext and the key used for the cipher

Lecture Handouts

Reading List