Workshop Lecture 3: Introduction to Cryptography

Joachim Rosenthal (University of Notre Dame)

Cryptography has a long history and its main objective is the transmission of data between two parties in a way which guarantees the privacy of the information. There are other interesting applications such as digital signatures, the problem of authentication and the concept of digital cash. The proliferation of computer networks resulted in a large demand for cryptography from the private sector.

In this lecture we will provide an introduction into the algebraic theory of cryptography. We will explain the difference between secret key cryptography and public key cryptography. In the area of secret key cryptography we will explain the new standard, the Rijndael system. All operations for this system can be explained through algebraic manipulations in a polynomial ring modulo a zero dimensional ideal.

We explain the Diffie-Hellman key exchange and the important concept of a one-way trapdoor function and we show how one-way trapdoor functions are built in practice.