Course detail

Geometrical Algorithms and Cryptography

FSI-SAVAcad. year: 2023/2024

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


Number of ECTS credits


Mode of study

Not applicable.

Entry knowledge

Basics of algebra. The craft of algoritmization.

Rules for evaluation and completion of the course

Exam: oral
Lectures: recommended


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

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

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-MAI-P Master's, 2. year of study, summer semester, compulsory-optional

Type of course unit



26 hours, optionally

Teacher / Lecturer


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.