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Discrete Mathematics

Module name (EN):
Name of module in study programme. It should be precise and clear.
Discrete Mathematics
Degree programme:
Study Programme with validity of corresponding study regulations containing this module.
Computer Science and Communication Systems, Master, ASPO 01.04.2016
Module code: KI873
Hours per semester week / Teaching method:
The count of hours per week is a combination of lecture (V for German Vorlesung), exercise (U for Übung), practice (P) oder project (PA). For example a course of the form 2V+2U has 2 hours of lecture and 2 hours of exercise per week.
3V+1U (4 hours per week)
ECTS credits:
European Credit Transfer System. Points for successful completion of a course. Each ECTS point represents a workload of 30 hours.
6
Semester: 2
Mandatory course: no
Language of instruction:
German
Assessment:
Exam

[updated 30.06.2024]
Applicability / Curricular relevance:
All study programs (with year of the version of study regulations) containing the course.

DFI-DM (P610-0269) Computer Science, Master, ASPO 01.10.2018 , semester 1, optional course
KI873 Computer Science and Communication Systems, Master, ASPO 01.04.2016 , semester 2, optional course, informatics specific
KIM-DM (P222-0051) Computer Science and Communication Systems, Master, ASPO 01.10.2017 , semester 1, mandatory course
PIM-DM (P222-0051) Applied Informatics, Master, ASPO 01.10.2011 , semester 2, mandatory course
PIM-DM (P222-0051) Applied Informatics, Master, ASPO 01.10.2017 , semester 1, mandatory course
Workload:
Workload of student for successfully completing the course. Each ECTS credit represents 30 working hours. These are the combined effort of face-to-face time, post-processing the subject of the lecture, exercises and preparation for the exam.

The total workload is distributed on the semester (01.04.-30.09. during the summer term, 01.10.-31.03. during the winter term).
60 class hours (= 45 clock hours) over a 15-week period.
The total student study time is 180 hours (equivalent to 6 ECTS credits).
There are therefore 135 hours available for class preparation and follow-up work and exam preparation.
Recommended prerequisites (modules):
None.
Recommended as prerequisite for:
Module coordinator:
Prof. Dr. Peter Birkner
Lecturer: Prof. Dr. Peter Birkner

[updated 31.03.2012]
Learning outcomes:
 
After successfully completing this module, students will have improved their knowledge about the concept of divisibility in the area of whole numbers. They will be able to recognize and apply divisibility relationships. Students will have worked with the lecturer to derive the congruence relation from divisibility. They will be familiar with the concept of the remainder class and will be able to calculate its inverse. They will be able to analyze the structure of residue class groups.
 
Students will be familiar with the Chinese Remainder Theorem. They will be able to derive the general proof from the proof for 2 equations. They will be able to apply the Chinese Remainder Theorem to specific tasks and use it to solve practical problems.
 
Students will be able to explain what prime numbers are. They will be able to estimate the number of prime numbers below a given threshold. They will be able to use a primality test to check whether a natural number is prime or not. They will be able to recognize pseudoprimes and know what this means for the primality test.
 
They will have improved their knowledge about the group theory. They will be familiar with various properties and structures, such as order, (cyclic) subgroup, generator, etc. They will be able to recognize these structures and apply them in different contexts.
 
Students will be able to identify the problem of the discrete logarithm. They will be able to solve it independently using the baby-step giant-step algorithm and perform the Diffie-Hellman protocol as an application.
 
The students will be able to explain what an elliptic curve is and know how to add points to it. They will be able to recognize the group structure in the set of points and apply both the field theory and the Diffie-Hellman protocol to elliptic curves.


[updated 30.06.2024]
Module content:
 
1. Module arithmetics
Divisibility, congruences, efficient modular exponentiation mod p, divisibility rules, residue classes, inverse residue classes, residue class groups, Euler´s phi function and its calculation
 
2. The Chinese Remainder Theorem (CRT)
CRT for 2 equations, CRT in general, examples and applications
 
3. Prime numbers
Prime numbers, fundamental theorem of algebra, there are infinitely many prime numbers, prime number theorem, Fermat´s little theorem, Fermat primality test, pseudoprimes
 
4. Group theory
Group axioms, subgroups, exponentiation in groups, cyclic groups, ordering of elements and groups, homomorphisms, kernel and image
 
5. The discrete logarithm (DL)
The DL, Square and Multiply Method, Shanks’ Baby-Step Giant-Step Algorithm, the Diffie-Hellman-Protocol
 
6. Field theory
Finite bodies, characteristics
 
7. Elliptic curves (EC)
EC, points on the EC, Weierstrass equation, group law, graphical addition, discriminant, number of points of an EC over F_p, the Hasse-Weil interval


[updated 30.06.2024]
Recommended or required reading:
 
- Ziegenbalg: Elementare Zahlentheorie (Beispiele, Geschichte, Algorithmen) Springer, 2015
 
- Washington: Elliptic Curves (Number Theory and Cryptography), Chapman& Hall, 2008
 
- Iwanowski, Lang: Diskrete Mathematik mit Grundlagen (Lehrbuch für Studierende von MINT-Fächern), Springer, 2014


[updated 30.06.2024]
Module offered in:
WS 2018/19, WS 2017/18, SS 2017, SS 2016, SS 2015, ... 
[Sun Dec 22 16:18:30 CET 2024, CKEY=pdm, BKEY=kim, CID=KI873, LANGUAGE=en, DATE=22.12.2024]