Geometrical Algorithms and Cryptography (FSI-SAV)

Academic year 2022/2023
Supervisor: doc. RNDr. Miroslav Kureš, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The convergence of mathematician and computer scientist points of view.
Learning outcomes and competences:
The algoritmization of some geometric and cryptographic problems.
Prerequisites:
Basics of algebra. The craft of algoritmization.
Course contents:
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.
Teaching methods and criteria:
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
Controlled participation in lessons:
Lectures: recommended
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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.
Literature - fundamental:
1. Bump, D., Algebraic Geometry, World Scientific 1998
2. Webster, R., Convexity, Oxford Science Publications, 1994
3. Bernstein, D., Buchmann, J., Dahmen, E., Post-Quantum Cryptography, Springer, 2009
4. Senechal., M., Quasicrystals and Geometry, Cambridge University Press, 1995
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
CŽV full-time study CZV Bases of Mechanical Engineering -- Ex 3 Compulsory-optional 1 1 S
N-MAI-P full-time study --- no specialisation -- Ex 3 Compulsory-optional 2 2 S