Course detail

Geometrical Algorithms and Cryptography

FSI-SAV-AAcad. year: 2025/2026

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.

Language of instruction

English

Number of ECTS credits

3

Mode of study

Not applicable.

Guarantor

doc. RNDr. Miroslav Kureš, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Basics of algebra. The craft of algoritmization.

Rules for evaluation and completion of the course

Exam: oral
Lectures: recommended

Aims

The convergence of mathematician and computer scientist points of view.
The algoritmization of some geometric and cryptographic problems.

Study aids

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)
Webster, R., Convexity, Oxford Science Publications, 1994 (EN)

Recommended reading

Kureš, Miroslav: Geometrické algoritmy (rukopis, příprava k tisku)

Classification of course in study plans

  • Programme N-AIM-A Master's 2 year of study, summer semester, elective
  • Programme N-MAI-A Master's 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. 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.