
MA522
Computer Algebra Fall 2021 Online on Zoom live or in Dabney 331, Tue&Thur 11:45am1:00pm 
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Peoples' home pages: Erich Kaltofen, Maple programs for the course.


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

1. Aug 17 
Administrative meeting.
Algorithm Defined.
Undecidability of the Halting Problem.

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


2. Aug 19 
First algorithms:
Karatsuba's Algorithm;
bigOh notation


GG §4.3, GG §20


3. Aug 24 
Algebraic Random Access Machine (RAM) model of computation;
analysis of Karatsuba's Algorithm;
bit complexity

Ka88_jacm.pdf, Section 2.

GG §4.3, GG §20


4. Aug 26 
Repeated squaring, RSA

3.mws,

GG §2; §3; §5.4


5. Aug 31 
Extended Euclidean algorithm;
Hermite elimination

hermite.mws,
lagrange.mws

GG §3.3; §4.5; §5.2


6. Sep 2 
Analysis of Euclid; Chinese remaindering

chrem.mws



Mon, Sep 6  Labor Day, no class  
7. Sep 7 
Newton and Lagrange interpolation;
ReedSolomon code and decoding

ISSAC 21 talk,
welch_berlekamp.mws

GG §5.10, §5.11


8. Sep 9 
Distribution of primes; use of interpolation/CRA;
rational number recovery; continued fraction approximations of a rational number

[Kaltofen and Villard
2004, p. 112] ,
[Kaltofen and Rolletschek 1989,
Theorem 5.1], KR_ratrec.mpl, KR_ratrec.mws,
cont_frac.pdf,
cont_frac.mws



9. Sep 14 
ReedSolomon decoding by rational function recovery

BW_rat_fun.mws,
notes.pdf

GG §5.8


10. Sep 16 
Pollard rho; birthday paradox

Pollard rho code: new_pollard_rho.mpl,
new_pollard_rho.mws,
new_pollard_rho.pdf

GG §19.4


11. Sep 21 
Primitive elements modulo p;
computing discrete logs via Shanks's babysteps/giant steps method and Pollard rho

Teske's paper;
discrete log Pollard rho code:
new_discrete_log.mpl



12. Sep 23 
Diffie/Hellman/Merkle key exchange; el Gamal crypto system


GG §20.3, 20.4


13. Sep 28 




14. Sep 30 




MonTue, Oct 45  Fall Break, no class  
15. Oct 7 
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 12 
Fractionfree Gaussian elimination




Wed, Oct 13, 23h59  Last day to drop the course 

17. Oct 14 
Fundamental theorem on subresultants


GG §6.10 and §11.2


18. Oct 19 
Unique factorization domains




19. Oct 21 
Algebraic extension fields; construction of a splitting field.



20. Oct 26 
Isomorphism of splitting fields; Galois group; separable and inseparable extensions




21. Oct 28 
Norms and traces; the fundamental theorem on symmetric functions;
the ring of algebraic integers


tower_of_fields.mws, tower_of_fields.txt.


22. Nov 2 
Cyclotomic extensions; the infrastructure of finite fields




23. Nov 4 
Factoring polynomials over finite fields:
the Berlekamp polynomial factoring algorithm; Camion's large primes method


GG §14


Fri, Nov 5  Topic for class presentation must be declared at 17h  
24. Nov 9 
Factoring polynomials over finite fields cont.:
the distinct degree and CantorZassenhaus algorithm

factmodp.mws

GG §14.8


Wed Nov 10  Approvals of topics for term papers by me are posted  
25. Nov 11 
Polynomial ideals; term orders; reduction




26. Nov 16 
Gröbner bases; Buchberger's algorithm


GG §21


27. Nov 18 
Buchberger's algorithm continued

groebner.mws



28. Nov 23 
Critical pair/completion paradigm: GCDfree basis construction

[Kaltofen 85, Section 3]



WednesdayFriday, Nov 2426 🦃  Thanksgiving, no class  
Thur., Dec 2, noon3:30pm  Presentations  
Friday, December 10, 11:59am  Fall grades due 
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 six homework assignments of approximately equal weight and a choice of a Maple programming project or a term paper. 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 or term paper  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, 2021 Erich Kaltofen. Permission to use provided that copyright notice is not removed.