Geometrical Algorithms and Cryptography
FSI-SAVAcad. year: 2020/2021
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.
Learning outcomes of the course unit
The algoritmization of some geometric and cryptographic problems.
Basics of algebra. The craft of algoritmization.
Recommended optional programme components
Recommended or required reading
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)
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
Language of instruction
The convergence of mathematician and computer scientist points of view.
Specification of controlled education, way of implementation and compensation for absences
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.