Computer Algebra (Ma 810a - Cs 810a)

Tuesdays and Thursdays from 05:00PM to 06:15PM

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Course Description:

• The course is designed for advanced undergraduate, graduate students in computer science and mathematics. It introduces basic topics and methods in computer algebra with emphasis on applications to cryptography. Topics include: fundamental algorithms for integers and polynomials, Euclidean algorithm, Chinese Remainder, Fast Fourier Transform, polynomial factorization and root finding, lattice basis reduction, primality testing, integer factorization, introduction to polynomial ideals and Gröbner basis. Programming skills are required as students are expected to complete a number of programming projects.

Pre-requisites:

• Programming skills in either C/C++, Matlab or Mathematica

Textbooks:

• Modern Computer Algebra (2nd edition)
  by Joachim von zur Gathen and Jürgen Gerhard, Cambridge University Press (2003), ISBN 0521826462.

Instructors:

• Dr. Alex Myasnikov,
  Office hours: By appointment. Pierce 308
• Dr. Werner Backes,
  Office hours: By appointment. Babbio 640b

Grading:

• to be decided

Policies:

• to be decided

Syllabus (prelimenary):

1. Introduction. Basics of algorithm analysis and complexity theory
   (definition of an algorithm, O(.) notation, polynomial and exponential time complexity)
2. Representation and addition of numbers and polynomials.
3. Multiplication, division with reminder.
4. Euclidean algorithm. The Extended Euclidean algorithm
5. Modular arithmetic. Repeated squaring.
6. Linear diophantine equations
7. Change of representation. Chinese Remainder Algorithm.
8. Fast Fourier Transform. Fast multiplication.
9. Newton iteration.
10. Solving polynomial equations using Newton iteration.
11. Fast polynomial evaluation and interpolation (maybe omit this one)
12. Fast Matrix multiplication (maybe omit this one)
13. Algorithms for polynomial factorization, root finding.
14. LLL basis reduction algorithm for lattices
15. Factoring polynomials using using basis reduction
16. Cryptanalysis of knapsack-type cryptosystems.
17. Primality testing.
18. Finding primes.
19. Integer factorization. Trial division.
20. Integer factorization algorithms.
21. Public key cryptography. RSA, Diffie-Hellman key exchange.
22. Polynomial ideals.
23. Gröbner bases and S-polynomials
24. Buchberger's algorithm (not sure if this can be done)
25. Multivariate cryptography.