##### Department of Mathematics,

University of California San Diego

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### Math 295 - Colloquium

## Joachim Rosenthal

#### University of Notre Dame

## Algebraic Methods in Cryptography

##### Abstract:

Modern cryptography is very algebraic by nature. In this overview talk we will explain the major secret and public key cryptographic protocols. For secret key systems, Rijndael has become the new standard and we will describe this protocol through a sequence of algebraic operations in a finite ring $R$. In the area of public key cryptography the major protocols are the RSA protocol, the traditional Diffie-Hellman and the ElGamal protocol. The last two protocols are based on the hardness of the discrete logarithm problem in a finite group. The discrete logarithm problem can be viewed as a semi-group action on a set. This leads naturally to a generalized Diffie-Hellman key exchange and a generalized ElGamal one-way trapdoor function. Using this point of view we will provide several interesting semi-group actions on finite sets. Our main focus will be examples of semi-ring actions on a semi-module. These examples may lead to practical new one-way trapdoor functions. The presented results constitute joint work with Gerard Maze and Chris Monico.

Host: Bill Helton

### February 5, 2004

### 3:00 PM

### AP&M 6438

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