Course detail

Geometrical Algorithms and Cryptography

FSI-SAVCompulsory-optionalMaster's (2nd cycle)Acad. year: 2016/2017Summer semester2. year of study4  credits

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.

Learning outcomes of the course unit

The algoritmization of some geometric and cryptographic problems.

Mode of delivery

90 % face-to-face, 10 % distance learning


Basics of algebra. The craft of algoritmization.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Bump, D., Algebraic Geometry, World Scientific 1998
Kureš, Miroslav: Geometrické algoritmy (rukopis, příprava k tisku)
Webster, R., Convexity, Oxford Science Publications, 1994
Bernstein, D., Buchmann, J., Dahmen, E., Post-Quantum Cryptography, Springer, 2009

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

Language of instruction


Work placements

Not applicable.


The convergence of mathematician and computer scientist points of view.

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

Lectures: recommended

Type of course unit



26 hours, optionally

Teacher / Lecturer


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.