Geometrical Algorithms and Cryptography (FSI-SAV-A)

Academic year 2018/2019
Supervisor: doc. RNDr. Miroslav Kureš, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
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 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. Convexity in euclidean spaces.
2. Voronoi diagrams.
3. Geodesic spaces.
4. Rings and fields.
5. Ideals and factorizations.
6. Polynomials, the ordering of polynomials.
7. Groebner basis.
8. Polynomial automorphisms.
9. Algebraic varieties, implicitization.
10. Elliptic and hyperelliptic curves.
11. Principles of asymmetric cryptography.
12. Cryptography based on elliptic curves.
13. Multivariate cryptosystems.
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
Literature - recommended:
1. Kureš, Miroslav: Geometrické algoritmy (rukopis, příprava k tisku)
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-A full-time study M-MAI Mathematical Engineering -- GCr 4 Compulsory 2 2 S