Course detail
Geometrical Algorithms and Cryptography
FSI-SAV 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
Czech
Number of ECTS credits
3
Supervisor
Department
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-MAI-P: 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.