### Discrete Mathematics 2

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

#### Lecturers

#### 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

##### University undergraduate

[FER3-EN] Computing - study

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

Elective Courses
(6. semester)
#### Learning Outcomes

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

#### Forms of Teaching

**Lectures**ex catedra, discussion with students

**Independent assignments**homework

**Laboratory**homerwork

#### Grading Method

Continuous Assessment | Exam | |||||
---|---|---|---|---|---|---|

Type | Threshold | Percent of Grade | Threshold | Percent of Grade | ||

Mid Term Exam: Written | 0 % | 50 % | 0 % | |||

Final Exam: Written | 0 % | 50 % |

#### Week by Week Schedule

- The Euclidean algorithm, Prime numbers
- Linear congruences; The Chinese Remainder Theorem
- Euler's phi-function
- Primitive roots; Solving some polynomial congruences
- The Legendre symbol, The Jacobi symbol
- The Quadratic Reciprocity Law
- Linear Diophantine equations, Pythagorean triples, Pell's equation
- Midterm exam
- Semigroups and groups
- Rings and fields
- Finite fields
- Introduction to cryptography
- Symmetric cryptography
- The RSA cryptosystem; Public-key cryptography
- 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*,#### For students

#### 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

#### Grading System

**85**Excellent

**70**Very Good

**55**Good

**45**Sufficient