Detailed Syllabus for
Cryptography – Mathematics 1580
Brown University – Fall, 2007
Professor Joe Silverman

1. An Introduction to Cryptography
   • Substitution ciphers
   • Mathematical preliminaries (divisibility, modular arithmetic, finite fields, counting arguments)
   • Symmetric and asymmetric ciphers
2. Discrete Logarithms and Diffie-Hellman
   • Historical interlude: Diffie, Hellman, Rivest, Shamir, and Adelman
   • The discrete logarithm problem
   • Diffie-Hellman key exchange
   • The ElGamal public key cryptosystem
3. Integer Factorization and RSA
   • More math (Euler's theorem, roots modulo pq)
   • The RSA public key cryptosystem
   • Primality testing
   • Smooth numbers and sieves
4. Probability Theory and Information Theory
   • Permuations, combinations, and probability theory
   • Collision algorithms and the birthday paradox
   • Pollard's ρ method
5. Elliptic Curves and Cryptography
   • Elliptic curves
   • Elliptic curves over finite fields and ECDLP
   • Elliptic curve cryptography
6. Digital Signatures
   • RSA digital signatures
   • Discrete log digital signatures
7. Lattices and Cryptography [as time permits]
   • Math review (vector spaces and linear algebra)
   • Integer lattices and the shortest vector problem
   • Applications of lattice reduction to cryptanalysis
   • Polynomial rings, quotient rings, and convolutions
   • The NTRU public key cryptosystem

Go to the Math 1580 Home Page.

Go to Professor Silverman's Home Page.