311.265 (21W) Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography)

Wintersemester 2021/22

Registration deadline has expired.

First course session
04.10.2021 15:00 - 17:00 HS 6 On Campus
... no further dates known

Overview

Due to the COVID-19 pandemic, it may be necessary to make changes to courses and examinations at short notice (e.g. cancellation of attendance-based courses and switching to online examinations).

For further information regarding teaching on campus, please visit: https://www.aau.at/en/corona.
Lecturer
Course title german Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography)
Type Lecture
Course model Attendance-based course
Hours per Week 2.0
ECTS credits 3.0
Registrations 19
Organisational unit
Language of instruction Englisch
possible language(s) of the assessment German , English , Spanish
Course begins on 04.10.2021
eLearning Go to Moodle course

Time and place

Please note that the currently displayed dates may be subject to change due to COVID-19 measures.
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Course Information

Intended learning outcomes

Special polynomial functions play an essential role in modern cryptography. The participants of this course will be familiarized with properties of polynomial functions on finite fields and residue class rings. The discussed topics are important for applications and provide an insight into the current research in this field. The participants should be motivated to perform their own research in this area.

Teaching methodology

Elaboration and discussion of properties and applications of polynomial functions in cryptography. 

Course content

1. Polynomial functions in modern cryptography

2. Algebraic properties of power functions

3. Fixed points of the power functions on GF(q) and Z_n

4. Chebyshev polynomials, Dickson polynomials and Lucas sequences

5. Dickson polynomial permutations on GF(q) and Z_n

6. Fixed points of Dickson polynomial permutations

7. Primality tests by Dickson polynomials

8. Pseudoprimes and Carmichael numbers

9. RSA-cryptosystems with wrong keys

10. Chains of permutable polynomials, Conjecture of Schur

Prior knowledge expected

Basic facts from algebra and elementary number theory.

Literature

Original papers on the topics of the lecture will be given during the course.

Some of the topics of this course are discussed on a high level in the book by Lausch, H. and W. Nöbauer: Algebra of Polynomials. North Holland Publishing Company, Amsterdam, 1973. 

Examination information

Im Fall von online durchgeführten Prüfungen sind die Standards zu beachten, die die technischen Geräte der Studierenden erfüllen müssen, um an diesen Prüfungen teilnehmen zu können.

Modified examination information (exceptional COVID-19 provisions)

In consideration of the Covid-19 situation the first examination on January 31 will be online. If somebody wishes a written exam I will arrange this too. 

Examination methodology

Written exam at the end of the semester. Further dates in the following semester.

Examination topic(s)

Content of the course and examples similar to the exercises of the course. 

Assessment criteria / Standards of assessment for examinations

The questions of the exam will be weighted with points. For a positive result one has to reach at least 50% of all possible points. 

Grading scheme

Grade / Grade grading scheme

Position in the curriculum

  • Thematic Doctoral Programme Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (SKZ: ---, Version: 16W.1)
    • Subject: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (Compulsory subject)
      • Modeling-Analysis - Optimization of discrete, continuous and stochastic systems ( 0.0h XX / 0.0 ECTS)
        • 311.265 Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography) (2.0h VO / 3.0 ECTS)
  • Bachelorstudium Technische Mathematik (SKZ: 201, Version: 17W.1)
    • Subject: Diskrete Mathematik (Compulsory elective)
      • 10.8 Ausgewählte Kapitel der Diskreten Mathematik ( 2.0h VO / 3.0 ECTS)
        • 311.265 Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography) (2.0h VO / 3.0 ECTS)
          Absolvierung im 5., 6. Semester empfohlen
  • Bachelor's degree programme Technical Mathematics (SKZ: 201, Version: 12W.2)
    • Subject: Diskrete Mathematik (Compulsory elective)
      • Ausgewählte Kapitel der Diskreten Mathematik ( 3.0h VU / 5.0 ECTS)
        • 311.265 Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography) (2.0h VO / 3.0 ECTS)
  • Master's degree programme Mathematics (SKZ: 401, Version: 18W.1)
    • Subject: Discrete Mathematics (Compulsory elective)
      • 6.7 Selected Topics in Discrete Mathematics ( 2.0h VO / 3.0 ECTS)
        • 311.265 Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography) (2.0h VO / 3.0 ECTS)
  • Master's degree programme Technical Mathematics (SKZ: 401, Version: 13W.1)
    • Subject: Diskrete Mathematik (Compulsory elective)
      • Algebraic Methods in Discrete Optimization ( 3.0h VU / 5.0 ECTS)
        • 311.265 Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography) (2.0h VO / 3.0 ECTS)
  • Doctoral programme Doctoral programme in Technical Sciences (SKZ: 786, Version: 12W.4)
    • Subject: Studienleistungen gem. § 3 Abs. 2a des Curriculums (Compulsory subject)
      • Studienleistungen gem. § 3 Abs. 2a des Curriculums ( 16.0h XX / 32.0 ECTS)
        • 311.265 Selected Topics in Discrete Mathematics (Permutation Polynomials in Cryptography) (2.0h VO / 3.0 ECTS)

Equivalent courses for counting the examination attempts

Wintersemester 2018/19
  • 311.265 VO Ausgewählte Kapitel der Diskreten Mathematik (Permutationspolynome in der Kryptographie) (2.0h / 3.0ECTS)