Discrete Mathematics 2

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

Lecturers

Prof.
Mario Krnić
Prof.
Mario Osvin Pavčević

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

  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

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

  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
  6. The Quadratic Reciprocity Law
  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,

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
0 Physical education excercises

Grading System

85 Excellent
70 Very Good
55 Good
45 Sufficient

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