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Course detail

Geometrical Algorithms and Cryptography

Course unit code: FSI-SAV
Academic year: 2016/2017
Type of course unit: compulsory-optional
Level of course unit: Master's (2nd cycle)
Year of study: 2
Semester: summer
Number of ECTS credits:
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.
Course contents (annotation):
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.
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.
Course curriculum:
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:

Lecture: 26 hours, optionally
Teacher / Lecturer: doc. RNDr. Miroslav Kureš, Ph.D.
Syllabus: 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.

The study programmes with the given course