Universität Wien

562412 VO Computer algebra (2005S)

Computer algebra

0.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Details

Language: German

Lecturers

Classes (iCal) - next class is marked with N

  • Tuesday 08.03. 15:00 - 17:00 Seminarraum
  • Thursday 10.03. 15:00 - 17:00 Seminarraum
  • Tuesday 15.03. 15:00 - 17:00 Seminarraum
  • Thursday 17.03. 15:00 - 17:00 Seminarraum
  • Tuesday 05.04. 15:00 - 17:00 Seminarraum
  • Thursday 07.04. 15:00 - 17:00 Seminarraum
  • Friday 08.04. 11:00 - 13:00 Seminarraum
  • Tuesday 12.04. 15:00 - 17:00 Seminarraum
  • Thursday 14.04. 15:00 - 17:00 Seminarraum
  • Friday 15.04. 11:00 - 13:00 Seminarraum
  • Tuesday 19.04. 15:00 - 17:00 Seminarraum
  • Thursday 21.04. 15:00 - 17:00 Seminarraum
  • Friday 22.04. 11:00 - 13:00 Seminarraum
  • Tuesday 26.04. 15:00 - 17:00 Seminarraum
  • Thursday 28.04. 15:00 - 17:00 Seminarraum
  • Friday 29.04. 11:00 - 13:00 Seminarraum
  • Tuesday 03.05. 15:00 - 17:00 Seminarraum
  • Tuesday 10.05. 15:00 - 17:00 Seminarraum
  • Thursday 12.05. 15:00 - 17:00 Seminarraum
  • Thursday 19.05. 15:00 - 17:00 Seminarraum
  • Tuesday 24.05. 15:00 - 17:00 Seminarraum
  • Tuesday 31.05. 15:00 - 17:00 Seminarraum
  • Thursday 02.06. 15:00 - 17:00 Seminarraum
  • Tuesday 07.06. 15:00 - 17:00 Seminarraum
  • Thursday 09.06. 15:00 - 17:00 Seminarraum
  • Tuesday 14.06. 15:00 - 17:00 Seminarraum
  • Thursday 16.06. 15:00 - 17:00 Seminarraum
  • Tuesday 21.06. 15:00 - 17:00 Seminarraum
  • Thursday 23.06. 15:00 - 17:00 Seminarraum
  • Tuesday 28.06. 15:00 - 17:00 Seminarraum
  • Thursday 30.06. 15:00 - 17:00 Seminarraum

Information

Aims, contents and method of the course

Computer algebra is a more recent area, where mathematical tools and compuetr software are developed for the exact (and not numerical) solution of equations. The basic objects
of computer algebra are numbers and polynomials.
We concentrate on topics in algorithmic number theory and on Groebner bases to solve systems of polynomial equations. We will also treat CACs, in particular pari gp and reduce. There will be sessions in the computer room.
The syllabus is as follows:

1.) The Euclidean algorithm

- The classical Euclidean algorithm
- The extended Euclidean Algorithm
- Cost analysis
- Modular inverses, linear Diophantine equations
- The Chinese Remainder algorithm

2.) Factorization of polynomials

- Squarefree factorization
- Berlekamp's algorithm
- The iterated Frobenius algorithm
- Testing irreducibility
- The Hensel Lifting

3.) Primality Testing

- The Fermat test
- Strong pseudoprimality test
- Finding primes
- The Solovay-Strassen test
- The Miller-Rabin test
- The complexity of primality testing

4.) Faktorization of integers

- Pollard's rho-method
- Pollard's (p-1)-method
- Lenstra's elliptic curve method

5.) Public key Cryprography

- The RSA cryptosystem
- The Diffie-Hellman key exchange protocol
- The ElGamal cryptosystem
- Elliptic curves

6.) Groebner Bases

- Polynomial ideals
- Monomial ideals and Hilbert's basis theorem
- Buchberger's algorithm

Assessment and permitted materials

Minimum requirements and assessment criteria

Examination topics

siehe
http://www.mat.univie.ac.at/studentinfo/studienplan/Studienplan-Diplom3.html

Reading list

1.) von zur Gathen, Joachim; Gerhard, Jürgen:
Modern computer algebra. 1999.
2.) Forster, Otto: Algorithmische Zahlentheorie. 1996.
3.) Buchmann, Johannes A.: Introduction to cryptography. 2004.
4.) Sturmfels, Bernd: Solving systems of polynomial equations. 2002
5.) Cox, David; Little, John; O'Shea, Donal: Ideals,
Varieties and Algorithms. 1997.

Association in the course directory

Last modified: Mo 07.09.2020 15:48