3 Degree Plans – Mathematics for Information Technologies

Coordinated by: Department of Algebra
Study branch coordinator: prof. RNDr. Aleš Drápal, CSc., DSc.

Specialization of the programme Mathematics for Information Technologies

This programme allows the student to specialize in two directions.

1. Mathematics for information security. This direction is focused on deepening the theoretical knowledge of number theory, probability theory, theory of error-correcting codes, complexity theory, theory of elliptic curves, and computer algebra applied to some of these subjects. Attention is also given to practical aspects such as internet security, standards in cryptography, and legal aspects of data security.
2. Computer geometry. This direction deepens theoretical knowledge in various algebraic and geometric subjects together with their applications in geometry of computer vision and robotics, computer graphics and image processing, optimization methods and numerical linear algebra.

Choice of a specialization

The direction is selected in three subsequent steps:

 choice of the topic of Master thesis at the beginning of Year 1
 choice of elective courses
 choice of elective topics for the state final exam

Assumed knowledge

It is assumed that an incoming student of this branch has sufficient knowledge of the following topics and fields:

Linear algebra, real analysis, and probability theory.
Foundations of commutative and computer algebra (Galois theory, integral extensions, discrete Fourier transformation), modular arithmetic, multiplicative groups, finite fields, basic classes of error-correcting codes, and the group operations on elliptic curves.
Basics of theoretic cryptography and geometric modelling. Programming in C.

Should an incoming student not meet these entry requirements, the coordinator of the study programme may assign a method of acquiring the necessary knowledge and abilities, which may for example mean taking selected bachelor's courses, taking a reading course with an instructor, or following tutored independent study.

3.1 Obligatory Courses

Code Subject Credits Winter Summer
NMMB403 Computer Algebra 2   6 3/1 C+Ex
NMMB405 Complexity for Cryptography   6 4/0 Ex
NMMB407 Probability and Cryptography   6 4/0 Ex
NMMB409 Convex optimization   9 4/2 C+Ex
NSZZ023 Diploma Thesis I   6 0/4 C
NSZZ024 Diploma Thesis II   9 0/6 C
NSZZ025 Diploma Thesis III   15 0/10 C

3.2 Elective Courses

The elective courses for the specialization Mathematics for Information Security are denoted by (IS). The courses for specialization Computer Geometry are denoted by (CG). It is required to earn at least 45 credits from this group.

Code Subject Credits Winter Summer
NMMB401 Automata and Convolutional Codes (IS) 6 3/1 C+Ex
NMMB402 Numerical Algorithms (IS) 6 3/1 C+Ex
NMMB404 Cryptanalytic Attacks (IS) 6 3/1 C+Ex
NMMB501 Network Certification Security (IS) 5 2/2 C+Ex
NMAG436 Curves and Function Fields (IS) 6 4/0 Ex
NMMB431 Authentication Schemes (IS)* 3 2/0 Ex
NMMB436 Steganography and Digital Media (IS) 3 2/0 Ex
NMMB437 Legal Aspects of Data Protection (IS) 3 2/0 Ex
NMMB531 Number Field Sieve (IS) 3 2/0 Ex
NMMB532 Standards and Cryptography (IS) 3 2/0 Ex
NMMB533 Mathematical Software (IS) * 3 1/1 C+Ex
NMMB534 Quantum Information (IS) 6 3/1 C+Ex
NMMB538 Elliptic Curves and Cryptography (IS) 6 3/1 C+Ex
NMAG401 Algebraic Geometry (CG) 5 2/2 C+Ex
NMMB440 Geometry of Computer Vision (CG) 6 2/2 C+Ex
NMMB442 Geometric Problems in Robotics (CG) 6 2/2 C+Ex
NMAG563 Introduction to complexity of CSP (CG) 3 2/0 Ex
NMMB536 Optimization and Approximation CSP (CG) 6 2/2 C+Ex
NMNV531 Inverse Problems and Regularization (CG) 5 2/2 C+Ex
NMNV407 Matrix Iterative Methods 1 (CG) 6 4/0 Ex
NMNV438 Matrix Iterative Methods 2 (CG) 5 2/2 C+Ex
NMNV534 Numerical Optimization Methods (CG) 5 2/2 C+Ex
NMMB535 Compressed Sensing (CG) 6 2/2 C+Ex
NPGR013 Special Functions and Transformations in Image Processing (CG) 3 2/0 Ex
NPGR010 Computer Graphics III (CG) 6 2/2 C+Ex
NMMB433 Geometry for Computer Graphics (CG) 3 2/0 Ex
NPGR029 Variational methods in image processing (CG) 3 2/0 Ex
NMMB331 Boolean function (IS) 3 2/0 Ex
NMMB333 Introduction to data analysis (CG, IS) 5 2/2 C+Ex
NTIN104 Foundations of theoretical cryptography (IS) 5 2/1 C+Ex

*) These courses will not be taught any more.

3.3 State Final Exam

Requirements for taking the final exam

 Earning at least 120 credits during the course of the study.
 Completion of all obligatory courses prescribed by the study plan.
 Earning at least 45 credits by completion of elective courses.
 Submission of a completed Master's Thesis by the submission deadline.

Oral part of the state final exam

The oral part of the final exam consists of two subject areas. One question is asked from common subject area 1. Student chooses either two topic from among 2A, 2B, 2C in case of the specialization Mathematics for information security, or two topics from among 2D, 2E, 2F, 2G in case of specialization Computer geometry. One question is asked from every chosen topic.

1. Basic mathematical subjects.
Complexity classes and computational models, randomness and pseudorandomness, algorithms for algebraic structures, convex optimization.

2A. Information and Coding.
Classical and quantum information and its transfer. Consequences of quantum Fourier transform to cryptography. Convolution codes. Hidden a damaged information.

2B. Number theoretic algorithms.
Factorization: Pollard rho and Pollard p-1, CFRAC algorithm (including root approximation by chain fractions and solution of Pell equation), quadratic sieve ( including Tonelli-Shanks algorithm). Basic methods for discrete logarithm: Pohlig-Hellman, Baby steps-Giant steps and index calculus.

2C. Elliptical curves.
basic properties of algebraic function fields and their groups of divisors. Weierstrass normal form of elliptic curve - equivalence and derivation. Picard group and addition of points on elliptic curve. Morphisms, endomorphisms and isogeny. Applications in cryptography.

2D. Computer vision and robotics.
Mathematical model of perspective camera. Calculation of movement of calibrated camera from the pictures of unknown scene. 3D reconstruction from two images of unknown scene. Geometry of three calibrated cameras. Denavit-Hartenberg description of kinematics of manipulator. Inverse kinematic problem of 6-arm serial manipulator - formulation and solution. Calibration of parameters of manipulator - formulation and solution.

2E. Image processing and computer graphics.
Modelling of inverse problems, regularization methods, digitization of image, deblurring, edge detection, image registration, compression, image synthesis, compressed sensing, analytical, kinematic and differential geometry.

2F. Approximation and optimization.
Convex optimization problems, duality, Lagrange dual problem. Algorithms for convex optimization, interior point method. Constraint satisfaction problem, algebraic approach to dichotomy conjecture. Weighted constraint satisfaction problem. Examples of computational problems formulated on wCSP, algebraic theory. Solution of problems with extremely large input.

2G. Numerical linear algebra.
LU and Choleski decomposition of a matrix, least squares, Krylov subspaces, matrix iterative methods (Arnoldi, Lanczos, joint gradients, generalized method of minimal residuum), QR algorithm, regularized methods for inverse problems, numerical stability.

3.4 Recommended Course of Study

1st year

Code Subject Credits Winter Summer
NMMB405 Complexity for Cryptography   6 4/0 Ex
NMMB409 Convex optimization   9 4/2 C+Ex
NMMB403 Computer Algebra 2   6 3/1 C+Ex
NMMB407 Probability and Cryptography   6 4/0 Ex
NSZZ023 Diploma Thesis I   6 0/4 C
  Optional and Elective Courses   27    

2nd year

Code Subject Credits Winter Summer
NSZZ024 Diploma Thesis II   9 0/6 C
NSZZ025 Diploma Thesis III   15 0/10 C
  Optional and Elective Courses   36