Fundamentals of Cryptography

Passport

• Status of the title. The 01.03.02 curriculum for the 2025 intake has a course “Cryptography” in semester VII, but not “Fundamentals of Cryptography”.
• Workload of the confirmed course. 90 hours, 2 credits: 60 contact hours (30 lecture hours and 30 seminar hours) and 30 hours of independent work.
• Department/lecturer. The curriculum lists the specialization as “Discrete Mathematics”; no specific lecturer is published.
• Final assessment. The form of assessment should be taken from the current curriculum and timetable; it cannot be extracted reliably from the available text of the curriculum.

Contents

No separate program for “Fundamentals of Cryptography” as such was found. The department publishes a detailed but later program, “Cryptography on Elliptic Curves and Lattices”; it does not stand in for the core semester VII course, so below it is explicitly marked as related material.

The lecturer of the related program is A. A. Illarionov, D.Sc. (Phys.-Math.). The abstract says that the course covers public-key cryptosystems on elliptic curves and lattices, including post-quantum motivations. The document lists the following 24 topics:

1. Complexity of algorithms, polynomial and exponential algorithms, the \(P=NP\) problem.
2. Elliptic curves over \(\mathbb R\) and two descriptions of point addition.
3. Elliptic curves over an arbitrary field.
4. The order of the group of points over a finite prime field and Hasse’s theorem.
5. Elliptic-curve analogues of Diffie–Hellman and ElGamal.
6. Current digital signature standards of Russia and the USA.
7. Cryptographic strength of elliptic-curve schemes.
8. Constructing elliptic curves suitable for cryptography.
9. Projective coordinates and optimization of computations.
10. Directions for further development of elliptic-curve cryptography.
11. Lattices, basis and determinant.
12. Minkowski’s convex body theorem.
13. Gram–Schmidt orthogonalization.
14. \(\mu\)-reduced bases and their construction.
15. Successive minima of lattices and the Hermite constant.
16. Lagrange’s algorithm for an \(L\)-reduced basis.
17. Hermite’s algorithm for finding a short vector.
18. Algorithmic lattice problems used in cryptography.
19. The \(LLL\)-reduced basis and its properties.
20. The \(LLL\) algorithm.
21. The GGH cryptosystem.
22. The GGH digital signature.
23. The NTRU cipher.
24. The NTRUSign digital signature.

The document also gives the expected learning outcomes: to know the properties of elliptic curves, lattices and reduced bases, the Hermite and \(LLL\) algorithms, lattice problems and the GGH/NTRU systems; to be able to prove properties of these objects and to estimate the complexity and correctness of the algorithms; to have command of the basics of the geometry of numbers and the cryptographic applications of these objects. Ongoing assessment in the related program consists of homework assignments, a possible programming assignment and a mid-semester colloquium; no numerical weights or thresholds are given.

Grading

No public grading scheme with numerical rules was found.

Sources and currency

01.03.02 curriculum, 2025 intake, program of the related course, list of the department’s bachelor’s courses.