Course detail
Geometrical Algorithms and Cryptography
FSI-SAV Acad. year: 2021/2022 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
Czech
Number of ECTS credits
3
Supervisor
Department
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 M2A-P: Applied Sciences in Engineering, Master's
branch M-MAI: Mathematical Engineering, compulsory-optional
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
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.