Course detail

Geometrical Algorithms and Cryptography

FSI-SAV-A Acad. year: 2024/2025 Summer semester

Basic outline of the lattice theory in vector spaces, Voronoi tesselation, computational geometry, commutative algebra and algebraic geometry with the emphasis on convexity, Groebner basis, Buchbereger algorithm and implicitization. Elliptic curves in cryptography, multivariate cryptosystems.

Language of instruction

English

Number of ECTS credits

3

Entry knowledge

Basics of algebra. The craft of algoritmization.

Rules for evaluation and completion of the course

Exam: oral
Lectures: recommended

Aims

The convergence of mathematician and computer scientist points of view.
The algoritmization of some geometric and cryptographic problems.

The study programmes with the given course

Programme N-AIM-A: Applied and Interdisciplinary Mathematics, Master's, elective

Programme N-MAI-A: Mathematical Engineering, Master's, compulsory-optional

Type of course unit

 

Lecture

26 hours, optionally

Syllabus

1. Discrete sets in affine space.
2. Delone sets.
3. k-lattices, Gram matrix, dual lattice.
4. Orders of quaternion algebras.
5. Voronoi cells. Facet vectors.
6. Fedorov solids. Lattice problems.
7. Principles of asymmetric cryptography. RSA system.
8. Elliptic and hypereliptic curves. Elliptic curve cryptography.
9. Polynomial rings, polynomial automorphisms.
10. Gröbner bases. Multivariate cryptosystems.
11. Algebraic varieties, implicitization. Multivariate cryptosystems.
12. Convexity in Euclidean and pseudoeucleidic spaces.
13. Reserve.