Algorithmic Algebra and Number Theory
General Information
- Course Number : MA.3532
- Time: Dienstag 10:15-12:00,
Donnerstag 8:15-10:00
- Lecturer: Laura Ciobanu
- E-mail: laura.ciobanu@unifr.ch
GAP
GAP for Mac download.
GAP for Windows.
MATHEMATICA WORKSHOPS
HOMEWORK
Course description
The course will present key concepts and algorithms in the three areas
specified below. A significant number of examples and algorithms will
be analyzed with the use of software such as Mathematica and GAP. No
programming experience is necessary.
I. Computational Number Theory.
A short introduction to cryptography: RSA, The discrete log problem,
Diffie-Hellman, ElGamal. Elliptic curves: the Weierstrass normal form,
the group law, counting rational points, the Schoof Algorithm.
Supersingular elliptic curves. The discrete logarithm problem for
elliptic curves. The Babystep-Giantstep algorithm, Factoring algorithms
using elliptic curves.
II. Commutative algebra from a computational viewpoint.
Ideals, Varieties, Groebner Bases, Buchberger's Algorithm, Hilbert's
Basis Theorem, The Nullstellensatz.
III. Computation in finitely presented groups.
Tietze transformations, Coset enumeration, The Reidemeister-Schreier
algorithm, Decision Problems, Computation in finite permutation groups.
Literatur
- Elliptische Kurven in der Kryptographie, by Annette Werner
(Springer Lehrbuch, 2002)
- Rational Points on Elliptic Curves, by Joseph H. Silverman and
John Tate (Springer-Verlag, 1992) 2. The Arithmetic of Elliptic Curves,
by Joseph H. Silvermann (Graduate Texts in Mathematics 106,
Springer-Verlag, 1986)
- Ideals, Varieties and Algorithms: An Introduction to
Computational Algebraic Geometry and Commutative Algebra, by David Cox,
John Little and Donal O'Shea (Springer-Verlag, New York, 2nd edition,
1997)
- Handbook of Computational Group Theory, by Derek Holt, Bettina
Eick and Eamonn O'Brien (Chapman & Hall/CRC, 2005).
- Computation with Finitely Presented Groups, by Charles Sims
(Cambridge University Press, 1994)
- Modern Computer Algebra (second edition), by Joachim von zur
Gathen and Juergen Gerhard (Cambridge University Press, 2003)