### 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 optimalization | 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 |

#### 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 optimalization | 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 |