|
|
| Lecture |
Topic(s) |
Notes |
Book(s) |
| 1. Aug 24 |
Administrative meeting. First algorithms:
modulo n arithmetic.
|
|
GG §4.3, GG §20
|
| 2. Aug 29 |
Zhengfeng Yang gives his
ISSAC 2006 talk
|
Link
to reference and paper.
|
|
| 3. Aug 31 |
No class
|
|
|
| 4. Sep 5 |
Repeated squaring, RSA
|
|
GG §4.3, GG §20
|
| 5. Sep 7 |
Extended Euclidean algorithm;
Chinese remaindering theorem/algorithm
|
|
GG §2; §3; §5.4
|
| 6. Sep 12 |
Hermite elimination;
analysis of Euclid;
Newton and Lagrange interpolation;
|
|
GG §3.3; §4.5; §5.2
|
| 7. Sep 14 |
Distribution of primes;
use of interpolation/CRA.
|
[Kaltofen and Villard 2004, p. 112]
|
|
| 8. Sep 19 |
Rational number recovery;
continued fraction approximations of a rational number
|
[Kaltofen and
Rolletschek 1989, Theorem 5.1],
KR_ratrec.mpl,
KR_ratrec.mws
|
GG §5.10, §5.11
|
| 9. Sep 21 |
Pollard rho;
birthday paradox
|
Pollard rho code:
pollard_rho.mpl,
pollard_rho.mws
|
GG §19.4
|
| 10. Sep 26 |
Talk in joint numeric analysis and symbolic computation seminars
|
see 173. MAPsncintro.pdf and 174. MAPissacKYZ.pdf
linked at
BASE/ lectures/ lectures. html# mapgenova
|
| 11. Sep 28 |
Primitive elements modulo p; computing discrete logs
via baby-steps/giant steps method and Pollard rho
|
Teske's paper
|
|
| 12. Oct 3 |
Maple experiments of Pollard rho;
Diffie/Hellman key exchange,
el Gamal crypto system
|
discrete_log.mpl
|
GG §20.3 and §20.4
|
| Wed, Oct 4, 5pm |
Last day to drop the course
|
| 13. Oct 4 |
Fraction-free Gaussian elimination
|
|
|
| 14. Oct 10 |
Definition of intergral domain,
field of quotients; Euclidean algorithm
for polynomials over a field;
Sylvester resultants
|
sylvester.mws,
sylvester.txt.
|
GG §25.2, §25.3 and §6.3
|
| Thurs-Fri, Oct 12-13 |
Fall Break, no class |
| 15. Oct 17 |
Fundamental theorem on subresultants
|
|
GG §6.10 and §11.2
|
| 16. Oct 19 |
Unique factorization domains
|
|
|
| 17. Oct 24 |
Algebraic extension fields;
construction of a splitting field.
|
|
|
| 18. Oct 26 |
Isomorphism of splitting fields;
Galois group;
separable and inseparable extensions
|
|
|
| 19. Oct 31 |
The Berlekamp/Massey algorithm
|
|
|
| 20. Nov 2 |
Norms and traces;
the fundamental theorem on symmetric functions
|
|
|
| 21. Nov 7 |
The ring of algebraic integers;
cyclotomic extensions;
the infrastructure of finite fields
|
|
|
| 23. Nov 14 |
Factoring polynomials over finite fields:
the distinct degree and Cantor-Zassenhaus algorithm
|
|
GG §14
|
|
Wed, Nov 15
|
Topic for class presentation must be declared at 5pm |
| 24. Nov 16 |
CanZas continued
|
|
|
|
Mon, Nov 20
|
Approvals of topics for term papers by me are posted |
| 25. Nov 21 |
The Berlekamp polynomial factoring algorithm;
Camion's large primes method
|
|
GG §14.8
|
| Thursday-Friday, Nov 22-24 |
Thanksgiving, no class |
| 26. Nov 28 |
Polynomial ideals; term orders;
reduction
|
|
GG §21
|
| 27. Nov 30 |
Gröbner bases; Buchberger's algorithm
|
|
|
| 28. Dec 5 |
Critical pair/completion paradigm: GCD-free basis construction
|
[Kaltofen 85, Section 3]
|
|
| 29. Dec 7 |
Wrap-up
|
|
|
|
Tue, Dec 12, 9-noon,
HA 335.
|
Presentations |
|
NEW
Requested/assigned times:
9-9:30: Nicole
9:30-10: Thomas
10-10:30: Robert
10:30-11: Rob
|
* This is a projected list and subject to amendment.
©2003 Erich Kaltofen. Permission to use
provided that copyright notice is not removed.
