Course detail

Geometrical Algorithms and Cryptography

FSI-SAV-A Acad. year: 2022/2023 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

Learning outcomes of the course unit

The algoritmization of some geometric and cryptographic problems.

Prerequisites

Basics of algebra. The craft of algoritmization.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Exam: oral

Aims

The convergence of mathematician and computer scientist points of view.

Specification of controlled education, way of implementation and compensation for absences

Lectures: recommended

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.