### Popis predmeta

#### 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.

#### 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

Independent assignments

Laboratory

#### 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

#### Study Programmes

Computing (study)
Elective Courses (6. semester)
Electrical Engineering and Information Technology (study)
Elective Courses (6. semester)
Audio Technologies and Electroacoustics (profile)
Free Elective Courses (2. semester)
Communication and Space Technologies (profile)
Free Elective Courses (2. semester)
Computational Modelling in Engineering (profile)
Free Elective Courses (2. semester)
Computer Engineering (profile)
Free Elective Courses (2. semester)
Computer Science (profile)
Free Elective Courses (2. semester)
Control Systems and Robotics (profile)
Free Elective Courses (2. semester)
Data Science (profile)
Free Elective Courses (2. semester)
Electrical Power Engineering (profile)
Free Elective Courses (2. semester)
Electric Machines, Drives and Automation (profile)
Free Elective Courses (2. semester)
Electronic and Computer Engineering (profile)
Free Elective Courses (2. semester)
Electronics (profile)
Free Elective Courses (2. semester)
Information and Communication Engineering (profile)
Free Elective Courses (2. semester)
Network Science (profile)
Free Elective Courses (2. semester)
Software Engineering and Information Systems (profile)
Free Elective Courses (2. semester)

#### 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 183494
Summer semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures
4 Laboratory exercises

Excellent
Very Good
Good
Acceptable

#### 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

Independent assignments

Laboratory

#### 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

#### Study Programmes

Computing (study)
Elective Courses (6. semester)
Electrical Engineering and Information Technology (study)
Elective Courses (6. semester)
Audio Technologies and Electroacoustics (profile)
Free Elective Courses (2. semester)
Communication and Space Technologies (profile)
Free Elective Courses (2. semester)
Computational Modelling in Engineering (profile)
Free Elective Courses (2. semester)
Computer Engineering (profile)
Free Elective Courses (2. semester)
Computer Science (profile)
Free Elective Courses (2. semester)
Control Systems and Robotics (profile)
Free Elective Courses (2. semester)
Data Science (profile)
Free Elective Courses (2. semester)
Electrical Power Engineering (profile)
Free Elective Courses (2. semester)
Electric Machines, Drives and Automation (profile)
Free Elective Courses (2. semester)
Electronic and Computer Engineering (profile)
Free Elective Courses (2. semester)
Electronics (profile)
Free Elective Courses (2. semester)
Information and Communication Engineering (profile)
Free Elective Courses (2. semester)
Network Science (profile)
Free Elective Courses (2. semester)
Software Engineering and Information Systems (profile)
Free Elective Courses (2. semester)

#### 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 183494
Summer semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures
4 Laboratory exercises