MA-522 Computer Algebra
Fall 2016
SAS 1220, Tue&Thu 11h55-13h10

Current Announcements

  • NEW Homework 2 is posted. Due Thurs. Dec 1, 5pm.
  • Homework 1 is posted.
  • I have further modified my Pollard rho implementations: the procedures pollard_rho in new_pollard_rho.mpl and discrete_log in new_discrete_log.mpl now return values derived from the collisions: factors and discrete logs. The procedure cycle_indices in new_pollard_rho.mpl is adjusted to process true/false returns from equal as well as [true,data] returned pairs. It assigns the data to a variable received on call. A difficulty is the recursive call discrete_log inside the equl procedure on a partially successful collision. Test runs are in the Maple worksheet new_pollard_rho.mws
  • The course web sites for Fall 2012, Fall 2009.
Old Announcements see below.

Peoples' home pages: Erich Kaltofen, Classlist

Maple programs for the course (Maple hints).

Homeworks

  • Homework 1 due Thurs Sep 29, 16h59, in my mailbox in SAS 3151.
  • Homework 2 due Thurs Dec 1, 16h59, in my mailbox in SAS 3151.
  • Homework 3, a 3-5 page term paper on your presentation.

Projects

Computer Help Resources

Syllabus

Course Outline*

Lecture Topic(s) Notes Book(s)
1. Aug 17 Administrative meeting. First algorithm: Freivalds's matrix multiplication verification by randomization


2. Aug 23 Integer and modulo n arithmetic; bit complexity

GG §4.3, GG §20
3. Aug 25 Repeated squaring, RSA
3.mws
GG §4.3, GG §20
4. Aug 30 Extended Euclidean algorithm; Chinese remaindering theorem/algorithm
4.mws, chrem.mws
GG §2; §3; §5.4
5. Sep 1 Hermite elimination; analysis of Euclid; Newton and Lagrange interpolation;
hermite.mws, lagrange.mws
GG §3.3; §4.5; §5.2
6. Sep 6 Distribution of primes; use of interpolation/CRA.
[Kaltofen and Villard 2004, p. 112]

7. Sep 8 Rational number recovery; continued fraction approximations of a rational number
[Kaltofen and Rolletschek 1989, Theorem 5.1], KR_ratrec.mpl, KR_ratrec.mws, 4.mws
GG §5.10, §5.11
8. Sep 13 More certificates in linear algebra: characteristic polynomial via crypto


9. Sep 15 Linearly recurrent sequences

GG §12.3
10. Sep 20 Sparse interpolation by the Prony-Blahut algorithm


11. Sep 22 Catch-up; Reed-Solomon decoding by rational function recovery
BW_rat_fun.mws
GG §5.8
12. Sep 27 Pollard rho; birthday paradox
Pollard rho code: new_pollard_rho.mpl, new_pollard_rho.mws
GG §19.4
13. Sep 29 Primitive elements modulo p; computing discrete logs via Shanks's baby-steps/giant steps method and Pollard rho
Teske's paper

14. Oct 4 Maple experiments of Pollard rho; Diffie/Hellman/Merkle key exchange, el Gamal crypto system; catch-up
new_discrete_log.mpl
GG §20.3 and §20.4
Thurs-Fri, Oct 6-7 Fall Break, no class
15. Oct 11 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
16. Oct 13 Fraction-free Gaussian elimination


Mon, Oct 15, 23h59 Last day to drop the course
17. Oct 18 Fundamental theorem on subresultants

GG §6.10 and §11.2
18. Oct 20 Unique factorization domains


19. Oct 25 Algebraic extension fields; construction of a splitting field.


20. Oct 27 Isomorphism of splitting fields; Galois group; separable and inseparable extensions


21. Nov 1 Norms and traces; the fundamental theorem on symmetric functions; the ring of algebraic integers

tower_of_fields.mws, tower_of_fields.txt.
22. Nov 3 Cyclotomic extensions; the infrastructure of finite fields


23. Nov 8 Factoring polynomials over finite fields: the Berlekamp polynomial factoring algorithm; Camion's large primes method

GG §14
24. Nov 10 Factoring polynomials over finite fields cont.: the distinct degree and Cantor-Zassenhaus algorithm
factmodp.mws
GG §14.8
Tues, Nov 15 Topic for class presentation must be declared at 17h
25. Nov 15 Polynomial ideals; term orders; reduction


26. Nov 17 Gröbner bases; Buchberger's algorithm

GG §21
Mon, Nov 21 Approvals of topics for term papers by me are posted
27. Nov 22 Buchberger's algorithm continued
groebner.mws

Wednesday-Friday, Nov 23-25 Thanksgiving, no class
28. Nov 29 Critical pair/completion paradigm: GCD-free basis construction
[Kaltofen 85, Section 3]

29. Dec 1 Wrap-up; possible presentation


Tuesday, Dec 6, 9am-12:00, SAS 1220. Presentations
Requested/assigned times:
* This is a previous list and I plan to rearrange it shortly, adding sparse interpolation

Textbook and Notes

I will be closely following whose sections are marked in the above syllabus by GG.

On-line information: All information on courses that I teach (except individual grades) is now accessible via html browsers, which includes this syllabus. Click on my courses' page of my resume. You can also find information on courses that I have taught in the past.

Grading and General Information

Grading will be done with plus/minus refinement.

There will be three homework assignments of approximately equal weight and one Maple programming projects. At the end of the course, each student will give a 30 minute presentation on material from the book not covered by me. A choice of topics will be provided by me. Class attendance will not be monitored in any way. If you need assistance in any way, please let me know (see also the University's policy).

Academic Standards

Late submissions: All homeworks and projects must be submitted on time. The following penalties are given for (unexcused) late submissions:

If you need assistance in any way, please let me know (see also the University's policy).

Old Announcements

©2009, 2012, 2016 Erich Kaltofen. Permission to use provided that copyright notice is not removed.