Discrete Mathematics 2

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
  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

Study Programmes

University undergraduate
Elective Courses (6. semester)
Elective Courses (6. semester)
University graduate
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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 183494
  Summer semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures
0 Seminar
0 Exercises
4 Laboratory exercises
0 Project laboratory

Grading System

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Good
Sufficient