M 390 C Applied Number Theory

INSTRUCTOR: Felipe Voloch (RLM 9.122, ph.471-2674)

CLASS HOURS: Tue, Thu 12:30 -- 2:00 RLM 11.176

OFFICE HOURS: Wed 10:00 -- 11:30 or by appointment.

TEXTBOOK: There will be no textbook, but part of the material I plan to cover can be found in various books such as Welsh' Codes and Cryptography or Schroeder's Number Theory in Science and Communication(which is more elementary).

Other good references are van Lint Introduction to Coding Theory and Koblitz A Course in Number Theory and Cryptography . Koblitz has a new book Algebraic Aspects of Cryptography which I haven't seen yet.

SYLLABUS: The purpose of this course is to introduce students to applications of Number Theory to Cryptography and Error-correcting codes. These two topics address the problem of preserving data integrity during transmission or storage. The first deals with protecting information against malicious attacks and the second against interference due to noise. Security on the Internet is a hot topic and it is all based on interesting mathematics. The prerequisites will be kept to a minimum, but previous exposure to elementary number theory or algebraic structures would be helpful.

Students may also want to check out EE 379 K.

Topics to be covered:

Basic properties of integers. Prime numbers and unique factorization. Congruences, Theorems of Fermat and Euler, primitive roots.

Primality testing and factorization methods. Introduction to finite fields. Running time of algorithms.

Cryptography, basic notions. Public key cryptosystems. RSA. Hash functions. Discrete log cryptosystems.

Error Correcting Codes, Vector spaces over finite fields, Hamming norm, coding, decoding. Examples of codes. Hamming, Golay, cyclic, BCH, Reed-Solomon, etc.