Study

# Computing

## Discrete Mathematics 2

• Class 45
• Practice 4
• Independent work 101
Total 150

### Course title

Discrete Mathematics 2

Elective

183494

6

5

### Course objectives

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.

### Required reading

(.), 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

#### Minimal 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
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