Course detail

Geometrical Algorithms and Cryptography

FSI-SAVAcad. year: 2018/2019

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.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Miroslav Kureš, Ph.D.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

The algoritmization of some geometric and cryptographic problems.

Prerequisites

Basics of algebra. The craft of algoritmization.

Co-requisites

Not applicable.

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

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The convergence of mathematician and computer scientist points of view.

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

Lectures: recommended

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Bernstein, D., Buchmann, J., Dahmen, E., Post-Quantum Cryptography, Springer, 2009 (EN)
Bump, D., Algebraic Geometry, World Scientific 1998 (EN)
Senechal., M., Quasicrystals and Geometry, Cambridge University Press, 1995 (EN)
Webster, R., Convexity, Oxford Science Publications, 1994 (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-MAI , 2 year of study, summer semester, compulsory-optional

Type of course unit

 

Lecture

26 hod., 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.