MAT 401  Seminar: Mathematical Cryptography  

Fall 2025, TuTh 2:00-3:20pm Physics P125

Instructor: Ljudmila Kamenova

e-mail: kamenova@math.stonybrook.edu
Office: Math Tower 3-115.
Office hours: TuTh 1:00-2:00pm, advising hour: Th 11:00am-12noon in Math Tower 3-115.

Syllabus: The seminar topic this semester will be "Mathematical Cryptography". The term cryptography refers to a wide range of security issues in the transmission and safeguarding of information. Many applications of algebra and number theory have arisen as a result of the development of public key cryptography. We'll introduce polynomial algorithms and randomized algorithms. We'll also recall the Euclidean algorithm for polynomials, prove the Hilbert Basis Theorem, and the Nullstellensatz. We'll apply algebraic and number theory methods to study hidden monomial cryptosystems, and combinatorial-algebraic cryprosystems. If time permits, we can also talk about elliptic and hyperelliptic cryptosystems. This course satisfies the SBC SPK requirement; as such, students are required to give presentations. MAT401 and MAT402 may be repeated, since the topic changes every time. All students wishing to receive departmental Honors in Mathematics must take MAT401 or MAT402 at least once (among other requirements).

This seminar course is aimed primarily at third- and fourth-year students majoring in mathematics.

I will start by giving a few initial lectures, but after a week or two the students will start their in-class presentations. During the first week of classes topics will be assigned to students.

Prerequisites: Basic understanding of algebra or number theory as in MAT 311, or MAT 312, or MAT 313 or equivalent.

Required Text: There is no required textbook. We are going to use mostly Neal Koblitz's book "Algebraic Aspects of Cryptography" (you can find a PDF version of this book online).

Here are some further reading materials:

Grading: The grading will be based primarily on class participation (40%), the quality of the student's in-class presentations (30%), and a final term paper consisting of the written version of one of the presentations (30%). The term paper should be typed of length 5 to 10 pages. Not attending regularly will have a negative effect on the grade. Regular attendance is therefore recommended.

The final term papers are due in the last week of classes.

List of topics (Lecture number/ date/ topic/ sections from the book/ lecturer) :
  1. (Aug 26) Introduction: early history; the idea of public key cryptography and RSA cryptosystem (1.1-1.3) -- Ljudmila K.

  2. (Aug 28) Secret sharing, passwords, practical and impractical cryptosystems (1.4-1.7) -- Ljudmila K.

  3. (Sept 2) The Big-O notation and estimates (2.1-2.3) -- Ljudmila K.

  4. (Sept 4) Complexity of computations: P, NP, NP-Completeness, and others (2.4-2.7) -- Jinyang J.

  5. (Sept 9) Fields (3.1) -- Sang Woo Y.

  6. (Sept 11) Finite fields (3.2).

  7. (Sept 16) Polynomial rings and the Euclidean algorithm (3.3-3.4) -- William L.

  8. (Sept 18) Grobner bases (3.5) -- Aubrey C.

  9. (Sept 23) The Imai-Matsumoto cryptosystem (4.1) -- Furkan A.

  10. (Sept 25) Patarin's little dragon cryptosystem (4.2) -- Jessica L.

  11. (Sept 30) Other hidden monomial cryptosystems (4.3) -- Eric C.

  12. (Oct 2) Combinatorial-algebraic cryptosystems: history and Brassard's theorem (5.1-5.2) -- Jerry J.

  13. (Oct 7) Concrete combinatorial-algebraic systems and computations (5.3-5.4) -- Riki H.

  14. (Oct 9) Designing a secure system (5.5-5.7) -- Joshua L.

  15. (Oct 16) Elliptic curves (6.1) -- Henry G.

  16. (Oct 21) Elliptic curve cryptosystems (6.2) -- Matthew H.

  17. (Oct 23) Classical number theory problems via elliptic curves (6.3) -- Sidney P.

  18. (Oct 28) Conjectures on elliptic curves (6.4) -- Adam B.

  19. (Oct 30) Hyperelliptic curves (6.5) -- Alexei P.

  20. (Nov 4) Hyperelliptic cryptosystems (6.6) -- Eamon Z.

  21. (Nov 6) Introduction to hyperelliptic curves: definitions and properties (A.1) -- Sean A.

  22. (Nov 11) Polynomial and rational functions (A.2) -- Kevi C.

  23. (Nov 13) Zeros and Poles (A.3) -- Alexander P.

  24. (Nov 18) Divisors (A.4) -- Nick C.

  25. (Nov 20) Topic TBA -- Thanh D.

  26. (Nov 25) Catch-up on the material/ discussions. (Thanksgiving week)

  27. (Dec 2) Shor's algorithm -- Maddie B-R.

  28. (Dec 4) Zero knowledge proofs -- Blue H.


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