Fachbereich Mathematik

# Elliptic Curves and Cryptography

## PD Dr. habil. Jörg Zintl

Contact:

`joerg.zintl  math.uni-tuebingen.de`

### Timetable:

 Lecture/Vorlesung: We/Mi 8.15 -  9.45 online via Ilias Fr/Fr 8.15 -  9.45 online via Ilias Tutorials/Übungen: Tu/Di 16.15 - 17.45 hybrid (online via Ilias or N15)

### Content:

Data security is more and more becoming an important issue of everyday life. Keeping certain pieces of information private and secret is a central aspect of cryptography, but by far not the only one. As the example of online transactions via the internet shows, the authentification of the rightful owner of information is equally important, and so is the protection of data from manipulation.

The aim of the lecture is to discuss how mathematics can be applied to address these challenges. We will discuss certain basic concepts of cryptography together with some standard algorithms, as well as possible attacks on cryptosystems. Although issues of implementation will be mentioned from time to time, programming is not part of the lecture.

The main focus is on the underlying mathematical methods. We will review or introduce some elementary theories, like finite and cyclic groups, congruent numbers, vector spaces over finite fields and projective geometry. The latter opens up a road to state-of the-art cryptography, based on elliptic curves. We will use the last third of the lecture for a very elementary approach to the underlying geometric ideas without assuming any prerequisites from algebraic geometry.

Throughout we will see many instances, where secure cryptosystems relate to some very deep and beautiful mathematics, like the theorem of Fermat, the Riemann hypothesis or the Weil conjectures.

Here are some key words:

• symmetric / asymmetric cryptosystems
• public-key-cryprography
• finite and cyclic groups, discrete logarithms
• standard algorithms and attacks
• signatures and certificates
• prime numbers and factorization
• curves in projective space
• elliptic curves in cryptography
• counting points over finite fields

### Important Notes:

1. Please register for the lecture online on Ilias.
2. All participiants need to register for the tutorials until  Friday, 22 October 2021, 12:00 h online via:
3. The lectures start on Wednesday, 20 October 2021.
4. The tutorials start on Tuesday, 26 October 2021. Local participiants are encouraged to participate in person in lecture hall N15.
5. There will be handouts to the lecures beforehand on Ilias. There you will also find weekly exercises and video recordings of the lectures.
6. The lecture is part of the CIVIS-program. Lectures, exercises and handouts will be in English. Exams can be taken in English and in German. In case of difficulties with the registration for the lecture please don't hesitate to contact me by email.

t.b.a.

### Prerequisites:

groups, fields, rings, vector spaces, congruent numbers, finite fields (e.g. lectures: "Lineare Algebra 1" + "Algebraische Strukturen")

### Books:

There will be a handout available. Please be aware that this handout will not be complete and not free from errors, so it cannot replace taking notes of your own.

 A. Beutelspacher, J. Schwenk, K. Wolfenstetter: Moderne Verfahren in der Kryptographie, Springer Verlag I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Crptography, Cambridge University Press J. Silverman: The Arithmetic of Elliptic Curves, Springer Verlag

There will be more recommendations to come.

### Proof of Participation:

Successful completion of the course requires an oral exam. Only candidates which participate regularly and actively in the tutorials are admitted to examination.

Die Prüfungsleistung zur Vorlesung wird durch erfolgreiches Ablegen einer mündlichen Prüfung erworben. Voraussetzung zur Zulassung zur mündlichen Prüfung ist die regelmäßige und aktive Teilnahme an den Übungen.

### Dates of exams:

08.03.22

29.03.22

The exams will take place on campus in accordance with the infection-prevention rules of the university. If personal presence is not possible, arrangements for an exam via video conference will be made.

Please contact Mrs. Kabagema-Bilan (elena.kabagema-bilanuni-tuebingen.de) to register.