Course: Selected chapters from number theory

« Back
Course title Selected chapters from number theory
Course code KMT/VKT@
Organizational form of instruction Lecture + Seminar
Level of course Master
Year of study 2
Semester Winter
Number of ECTS credits 2
Language of instruction Czech
Status of course Compulsory
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Dofková Radka, doc. PhDr. Ph.D.
  • Pastor Karel, doc. Mgr. Ph.D.
  • Zdráhal Tomáš, doc. RNDr. CSc.
Course content
This course focuses on one of the key applications of number theory - the RSA encryption algorithm, which is a fundamental element of asymmetric cryptography. RSA is widely used for secure information exchange without the need to share a secret key. The security of this system lies in the difficulty of factoring large numbers, a problem in number theory. Course Content: 1. Introduction to Cryptography: Basic principles of cryptography Differences between symmetric and asymmetric cryptography History and significance of number theory in cryptography 2. Mathematical Foundations of RSA: Basics of number theory (prime numbers, divisibility, modular arithmetic) Fermat's Little Theorem and Euler's Theorem Principles of encryption and decryption in the RSA system 3. RSA Algorithm: Key generation: Selecting large prime numbers and calculating public and private keys Encryption and decryption process Practical examples of encrypting and decrypting numbers 4. RSA Security: Computational complexity of prime factorization Factoring large numbers and its practical implications Importance of key size for the security of the encryption system 5. Practical Implementation: Using mathematical software (e.g., Wolfram Mathematica) to implement RSA Importance of computational tools for modern cryptography Examples of encrypting real data using RSA Basic literature is available in the relevant team in MS Teams under the name Matematicka_podstata_metody_RSA.pdf.

Learning activities and teaching methods
unspecified
Learning outcomes
The aim of the course is to familiarize students with the principles and mathematical foundations of the RSA encryption algorithm, which is a key application of number theory in the field of asymmetric cryptography. Students will learn how RSA works, how to generate and use keys for encryption and decryption, and understand the importance of number theory for the security of digital communication.

Prerequisites
Prerequisites include basic knowledge of number divisibility (as covered in Algebra 1) and practical knowledge of the Wolfram Mathematica/Wolfram Cloud environment.

Assessment methods and criteria
unspecified
Specific requirements (including individual assignment of term papers) are listed in the relevant team in MS Teams. As for active physical participation, it is at least 49% for PS and KS students, at least 20% for students studying according to ISP (the requirements for ISP must be clarified with the teacher in person at the beginning of the semester).
Recommended literature
  • DERBYSHIRE J. (2007). Posedlost prvočísly. Praha.
  • DEVLIN K. (2005). Problémy pro třetí tisíciletí. Praha.
  • HALAŠ, R. (2014). Úvod do teorie čísel. Olomouc.
  • KŘÍŽEK M., SOMER L., ŠOLCOVÁ A. (2011). Kouzlo čísel. Praha.
  • SINGH S. (2000). Velká Fermatova věta. Praha.
  • STILLWELL, John. (2003). Elements of number theory. New York.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Education Study plan (Version): Teaching Mathematics for Lower Secondary Schools (NA23) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching mathematics at lower secondary schools (NA24) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching Mathematics for Lower Secondary Schools (NA22) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching Mathematics for Lower Secondary Schools (NA21) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching Mathematics for Lower Secondary Schools (NA20) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching mathematics at lower secondary schools (NA20) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching mathematics at lower secondary schools (NA23) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching Mathematics for Lower Secondary Schools (NA24) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching mathematics at lower secondary schools (NA22) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Education Study plan (Version): Teaching mathematics at lower secondary schools (NA21) Category: Pedagogy, teacher training and social care 2 Recommended year of study:2, Recommended semester: Winter