
MA522
Computer Algebra Fall 2018 Dabney 220, Mon&Wed 1:30pm2:45pm 
Syllabus  People  Maple  Projects  Homeworks  Reading  Grading  Academics 
Current Announcements

Peoples' home pages:
Erich Kaltofen,
Maple programs for the course (Maple hints).


Course Outline*  
Lecture  Topic(s)  Notes  Book(s)  

1. Aug 22 
Administrative meeting.
Algorithm Defined.

Robert McNaughton,
Elementary Computability, Formal Languages, and Automata,
Section 1.1.


2. Aug 24 
First algorithms: Freivalds's matrix multiplication verification by randomization,
integer and modulo n arithmetic.
Algebraic Random Access Machine (RAM) model of computation, bit complexity 
KA89_slpfac.pdf, Section 3.

GG §4.3, GG §20


3. Aug 27 
Repeated squaring, RSA

3.mws

GG §4.3, GG §20


4. Aug 29 
Extended Euclidean algorithm; Chinese remaindering theorem/algorithm

4.mws,
chrem.mws

GG §2; §3; §5.4


Mon, Sep 3  Labor Day, no class  
5. Sep 5 
Hermite elimination; analysis of Euclid; Newton and Lagrange interpolation;

hermite.mws,
lagrange.mws

GG §3.3; §4.5; §5.2


6. Sep 10 
Distribution of primes; use of interpolation/CRA.

[Kaltofen and Villard 2004, p. 112]



7. Sep 12 
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


Mon Sep 17, Wed Sep 19  No class
(I am at
ICERM)


8. Sep 24 
More certificates in linear algebra: characteristic
polynomial via crypto




9. Sep 26 
Linearly recurrent sequences


GG §12.3


10. Oct 1 
Sparse interpolation by the PronyBCH decoding algorithm




11. Oct 3 
Catchup; ReedSolomon decoding by rational function recovery

BW_rat_fun.mws

GG §5.8


12. Oct 8 
Pollard rho; birthday paradox

Pollard rho code: new_pollard_rho.mpl, new_pollard_rho.mws

GG §19.4


13. Oct 10 
Primitive elements modulo p;
computing discrete logs via Shanks's babysteps/giant steps method and Pollard rho

Teske's paper



ThursFri, Oct 45  Fall Break, no class  
14. Oct 15 
Maple experiments of Pollard rho;
Diffie/Hellman/Merkle key exchange, el Gamal crypto system;
catchup

new_discrete_log.mpl

GG §20.3 and §20.4


15. Oct 17 
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 22 
Fractionfree Gaussian elimination




17. Oct 24 
Fundamental theorem on subresultants


GG §6.10 and §11.2


Fri, Oct 19, 23h59  Last day to drop the course 

18. Oct 29 
Unique factorization domains




19. Oct 31 🎃 
Algebraic extension fields; construction of a splitting field.



20. Nov 5 
Isomorphism of splitting fields; Galois group; separable and inseparable extensions




21. Nov 7 
Norms and traces; the fundamental theorem on symmetric functions;
the ring of algebraic integers


tower_of_fields.mws, tower_of_fields.txt.


22. Nov 12 
Cyclotomic extensions; the infrastructure of finite fields




23. Nov 14 
Factoring polynomials over finite fields:
the Berlekamp polynomial factoring algorithm; Camion's large primes method


GG §14


Fri, Nov 16  Topic for class presentation must be declared at 17h  
24. Nov 19 
Factoring polynomials over finite fields cont.:
the distinct degree and CantorZassenhaus algorithm

factmodp.mws

GG §14.8


Tue, Nov 20  Approvals of topics for term papers by me are posted  
WednesdayFriday, Nov 2123  Thanksgiving, no class  
25. Nov 26 
Polynomial ideals; term orders; reduction




26. Nov 28 
Gröbner bases; Buchberger's algorithm


GG §21


27. Dec 3 
Buchberger's algorithm continued

groebner.mws



28. Dec 5 
Critical pair/completion paradigm: GCDfree basis construction

[Kaltofen 85, Section 3]



29. Dec 7 
Wrapup; possible presentation


Friday, Dec 14, 13pm16pm, Dabney 220  Presentations  
Requested/assigned times: 
Online 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.
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).
Grade split up  
Accumulated homework grade  40% 
Maple project  30% 
Presentation  30% 
Course grade  100% 
If you need assistance in any way, please let me know (see also the University's policy).
©2009, 2012, 2016, 2018 Erich Kaltofen. Permission to use provided that copyright notice is not removed.