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MA-522
Computer Algebra Fall 2018 Dabney 220, Mon&Wed 1:30pm-2:45pm |
Syllabus | People | Maple | Projects | Homeworks | Reading | Grading | Academics |
Current Announcements
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Peoples' home pages:
Erich Kaltofen,
Maple programs for the course (Maple hints).
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Course Outline* | |||||
Lecture | Topic(s) | Notes | Book(s) | ||
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1. Aug 22 |
Administrative meeting.
Algorithm Defined.
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Robert McNaughton,
Elementary Computability, Formal Languages, and Automata,
Section 1.1.
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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.
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GG §4.3, GG §20
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3. Aug 27 |
Repeated squaring, RSA
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3.mws
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GG §4.3, GG §20
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4. Aug 29 |
Extended Euclidean algorithm; Chinese remaindering theorem/algorithm
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4.mws,
chrem.mws
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GG §2; §3; §5.4
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Mon, Sep 3 | Labor Day, no class | ||||
5. Sep 5 |
Hermite elimination; analysis of Euclid; Newton and Lagrange interpolation;
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hermite.mws,
lagrange.mws
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GG §3.3; §4.5; §5.2
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6. Sep 10 |
Distribution of primes; use of interpolation/CRA.
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[Kaltofen and Villard 2004, p. 112]
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7. Sep 12 |
Rational number recovery; continued fraction approximations of a rational number
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[Kaltofen and Rolletschek 1989, Theorem 5.1], KR_ratrec.mpl, KR_ratrec.mws, 4.mws
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GG §5.10, §5.11
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Mon Sep 17, Wed Sep 19 | No class
(I am at
ICERM)
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8. Sep 24 |
More certificates in linear algebra: characteristic
polynomial via crypto
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9. Sep 26 |
Linearly recurrent sequences
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GG §12.3
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10. Oct 1 |
Sparse interpolation by the Prony-BCH decoding algorithm
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11. Oct 3 |
Catch-up; Reed-Solomon decoding by rational function recovery
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BW_rat_fun.mws
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GG §5.8
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12. Oct 8 |
Pollard rho; birthday paradox
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Pollard rho code: new_pollard_rho.mpl, new_pollard_rho.mws
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GG §19.4
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13. Oct 10 |
Primitive elements modulo p;
computing discrete logs via Shanks's baby-steps/giant steps method and Pollard rho
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Teske's paper
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Thurs-Fri, Oct 4-5 | Fall Break, no class | ||||
14. Oct 15 |
Maple experiments of Pollard rho;
Diffie/Hellman/Merkle key exchange, el Gamal crypto system;
catch-up
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new_discrete_log.mpl
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GG §20.3 and §20.4
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15. Oct 17 |
Definition of intergral domain, field of quotients; Euclidean algorithm for polynomials over a field; Sylvester resultants
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sylvester.mws, sylvester.txt.
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GG §25.2, §25.3 and §6.3
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16. Oct 22 |
Fraction-free Gaussian elimination
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17. Oct 24 |
Fundamental theorem on subresultants
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GG §6.10 and §11.2
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Fri, Oct 19, 23h59 | Last day to drop the course |
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18. Oct 29 |
Unique factorization domains
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19. Oct 31 🎃 |
Algebraic extension fields; construction of a splitting field.
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20. Nov 5 |
Isomorphism of splitting fields; Galois group; separable and inseparable extensions
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21. Nov 7 |
Norms and traces; the fundamental theorem on symmetric functions;
the ring of algebraic integers
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tower_of_fields.mws, tower_of_fields.txt.
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22. Nov 12 |
Cyclotomic extensions; the infrastructure of finite fields
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23. Nov 14 |
Factoring polynomials over finite fields:
the Berlekamp polynomial factoring algorithm; Camion's large primes method
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GG §14
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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 Cantor-Zassenhaus algorithm
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factmodp.mws
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GG §14.8
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Tue, Nov 20 | Approvals of topics for term papers by me are posted | ||||
Wednesday-Friday, Nov 21-23 | Thanksgiving, no class | ||||
25. Nov 26 |
Polynomial ideals; term orders; reduction
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26. Nov 28 |
Gröbner bases; Buchberger's algorithm
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GG §21
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27. Dec 3 |
Buchberger's algorithm continued
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groebner.mws
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28. Dec 5 |
Critical pair/completion paradigm: GCD-free basis construction
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[Kaltofen 85, Section 3]
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29. Dec 7 |
Wrap-up; possible presentation
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Friday, Dec 14, 13pm-16pm, Dabney 220 | Presentations | ||||
Requested/assigned times: |
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.
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.