Advanced topics in algebra and combinatorics from current problems of interest. May be repeated for credit, but only if different topics are covered.

Sample Offerings:

Introduction to Cryptography

The purpose of this course is to acquaint the students with classical and modern methods of cryptography and their uses in modern communication systems. Main topics: Shannon's theory, conventional cryptosystems, DES, AES, finite fields and elementary number theory, RSA, Diffie-Hellman key exchange scheme, ElGamal cryptosystem, digital signature schemes, elliptic curves and elliptic curve cryptosystems, hash functions, pseudorandom numbers, identification schemes, and zero knowledge proofs.

Coding Theory

This courses covers the basics of coding theory. Topics include cyclic codes, BCH codes, Reed-Solomon codes, and finite geometry.

Finite Fields

This course covers basic finite field theory and applications.

Algebraic Curves

This course covers some basic results about algebraic curves that are useful in constructing error-correcting codes and in implementing public-key cryptosystems. Basic concepts in algebraic geometry and commutative algebra to be covered include varieties, polynomial and rational maps, divisors, (prime) ideals, function fields, valuations, local rings, Riemann-Roch Theorem, etc.

Introduction to Computational Algebra I

The course focuses heavily on the theory and applications of Grobner bases. Coding theory is emphasized as an area of application, including decoding of Reed-Solomon codes and Hermitian codes.

Introduction to Computational Algebra II

Fast Fourier transforms, fast multiplication of polynomials (integers), fast decoding of RS codes, sparse linear systems (from coding theory, cryptography and computer algebra), Krylov subspace methods (Lanczos and bi-orthogonal methods), Wiedemann's method a la Berlekamp-Massey, block algorithms (Coppersmith's and Montgomory's) and their analysis.