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

NMMB403Computer Algebra 2 63/1 C+Ex
NMMB405Complexity for Cryptography 64/0 Ex
NMMB407Probability and Cryptography 64/0 Ex
NMMB409Convex optimalization 94/2 C+Ex
NSZZ023Diploma Thesis I 60/4 C
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 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.

NMMB401Automata and Convolutional Codes(IS)63/1 C+Ex
NMMB402Numerical Algorithms(IS)63/1 C+Ex
NMMB404Cryptanalytic Attacks(IS)63/1 C+Ex
NMMB501Network Certification Security(IS)52/2 C+Ex
NMAG436Curves and Function Fields(IS)64/0 Ex
NMMB431Authentication Schemes(IS)32/0 Ex
NMMB436Steganography and Digital Media(IS)32/0 Ex
NMMB437Legal Aspects of Data Protection(IS)32/0 Ex
NMMB531Number Field Sieve(IS)32/0 Ex
NMMB532Standards and Cryptography(IS)32/0 Ex
NMMB533Mathematical Software(IS)31/1 C+Ex
NMMB534Quantum Information(IS)63/1 C+Ex
NMMB538Elliptic Curves and Cryptography(IS)63/1 C+Ex
NMAG401Algebraic Geometry(CG)52/2 C+Ex
NMMB440Geometry of Computer Vision(CG)62/2 C+Ex
NMMB442Geometric Problems in Robotics(CG)62/2 C+Ex
NMAG563Introduction to complexity of CSP(CG)32/0 Ex
NMMB536Optimization and Approximation CSP(CG)62/2 C+Ex
NMNV531Inverse Problems and Regularization(CG)52/2 C+Ex
NMNV407Matrix Iterative Methods 1(CG)64/0 Ex
NMNV438Matrix Iterative Methods 2(CG)52/2 C+Ex
NMNV534Numerical Optimization Methods(CG)52/2 C+Ex
NMMB535Compressed Sensing(CG)62/2 C+Ex
NPGR013Special Functions and Transformations in Image Processing(CG)32/0 Ex
NPGR010Computer Graphics III(CG)62/2 C+Ex
NMMB433Geometry for Computer Graphics(CG)32/0 Ex
NPGR029Variational methods in image processing(CG)32/0 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

NMMB405Complexity for Cryptography 64/0 Ex
NMMB409Convex optimalization 94/2 C+Ex
NMMB403Computer Algebra 2 63/1 C+Ex
NMMB407Probability and Cryptography 64/0 Ex
NSZZ023Diploma Thesis I 60/4 C
 Optional and Elective Courses 27  

2nd year

NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 C
 Optional and Elective Courses 36  

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