### Discrete Mathematics 2

Data is displayed for academic year: 2023./2024.

#### Course Description

Euclidean algorithm. Linear congruences and systems. Euler phi function and prime roots. Quadratic residues. Pythagorean triples. Pell's equation. Groups, rings and fields. Public key cryptography.

#### Study Programmes

[FER3-EN] Computing - study
Elective Courses (6. semester)
[FER3-EN] Electrical Engineering and Information Technology - study
Elective Courses (6. semester)

#### Learning Outcomes

1. To solve linear congruence and a system of linear congruences.
2. Solve some of the polynomial and exponential congruences via prime roots.
3. Examine the solution existence of quadratic congruence by virtue of the Jacobi symbol.
4. Solve some basic diophantine equations.
5. Compute in finite fields.
6. Apply number theory and group theory in public key cryptography.

#### Forms of Teaching

Lectures

ex catedra, discussion with students

Independent assignments

homework

Laboratory

homerwork

Continuous Assessment Exam
Mid Term Exam: Written 0 % 50 % 0 %
Final Exam: Written 0 % 50 %

#### Week by Week Schedule

1. The Euclidean algorithm, Prime numbers
2. Linear congruences; The Chinese Remainder Theorem
3. Euler's phi-function
4. Primitive roots; Solving some polynomial congruences
5. The Legendre symbol, The Jacobi symbol
7. Linear Diophantine equations, Pythagorean triples, Pell's equation
8. Midterm exam
9. Semigroups and groups
10. Rings and fields
11. Finite fields
12. Introduction to cryptography
13. Symmetric cryptography
14. The RSA cryptosystem; Public-key cryptography
15. Final exam

#### Literature

(.), Andrej Dujella, Uvod u teoriju brojeva, https://web.math.pmf.unizg.hr/~duje/utb/utblink.pdf,
(.), K. H. Rosen: Elementary Number Theory and Its Applications, Addison-Wesley, Reading, 1993.,
(.), D. Žubrinić, Diskretna matematika, Element, 1997.,
(.), Course in Number Theory and Cryptography N. Koblitz Springer 1994,
(.), A. Baker: A Concise Introduction to the Theory of Numbers, Cambridge University Press, Cambridge, 1994.,
(.), I. Niven, H. S. Zuckerman, H. L. Montgomery: An Introduction to the Theory of Numbers, Wiley, New York, 1991.,
(.), A. Baker: A Comprehensive Course in Number Theory, Cambridge University Press, Cambridge, 2012.,
(.), Cryptography. Theory and Practice D. R. Stinson CRC Press 2002,

#### General

ID 223339
Summer semester
5 ECTS
L0 English Level
L1 e-Learning
45 Lectures
0 Seminar
0 Exercises
4 Laboratory exercises
0 Project laboratory