# Elliptic Curve Cryptography

Corresponding entry in Aachen Campus, Bonn University (Lecture, Tutorial).

## Responsible

Prof. Dr. Joachim von zur Gathen

## Lecture

## Tutorial

## Time & Place

- Tuesday 13
^{00}-14^{30}, b-it Rheinsaal. - Wednesday 13
^{00}-14^{30}, b-it Rheinsaal. - Tutorial: Tuesday 14
^{45}-16^{15}, b-it Rheinsaal or b-it 1.25.

**First meeting: Tuesday, 27 October 2009, 13 ^{30}, b-it Rheinsaal.**

Additional hours: Wednesday 14^{45}-15^{30}, b-it 1.25 on 2, 9, 16 December, 13, 20, 27 January and 3 February.

In exchange there is no course on 3, 22 and 23 December. The first course in 2010 is on 12 January.

## Exam

There will be oral exams. Dates are already set and sent to participants by email.

## Prerequisites

Basic knowledge in cryptography is required.

## Contents

We intend to introduce elliptic curves as used in cryptography. That encompasses definition and properties of the group law and of pairings. The necessary maths will be discussed in the course.

The computationally interesting part starts when we try to implement elliptic curve arithmetic as good as possible. The last few years have brought up quite a few surprising developments. One of those was the rise of Edwards curves. They are very close relatives of elliptic curves but more symmetric than the former in their shape and -more important- in their arithmetic (no special cases!).

All times subject to agreement in class.

## Notes

The screen notes (PDF) contain all handwritten stuff (last updated 28 January 2010, 18:20).

There is also a sketch (PDF) of the course (last updated 01 February 2010, 18:41).

## Exercises

- Exercise 1 (PDF),
- Exercise 2 (PDF),
- Exercise 3 (PDF),
- Exercise 4 (PDF),
- Exercise 5 (PDF),
- Exercise 6 (PDF),
- Exercise 7 (PDF),
- Exercise 8 (PDF),
- Exercise 9 (PDF),
- Exercise 10 (PDF),
- Exercise 11 (PDF).

## Literature

- Lawrence C. Washington (2003).
*Elliptic Curves — Number Theory and Cryptography*. Discrete Mathematics and its Applications. CRC Press, Boca Raton, FL, USA. ISBN 1-58488-365-0.

This is

*the*introduction to elliptic curves. The presentation only touches briefly the cryptographic situation but covers all mathematics. - Daniel Hankerson, Alfred Menezes & Scott Vanstone (2004).
*Guide to Elliptic Curve Cryptography*. Springer-Verlag, New York. ISBN 0-387-95273-X.

An introductory book covering the most important aspects: Arithmetic, cryptographic protocols, and implementation.

- Ian Blake, Gadiel Seroussi & Nigel Smart (1999).
*Elliptic Curves in Cryptography*. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press. ISBN 0-521-65374-6.

This book provides a steep introduction to ellitpic curves and all important aspects for cryptography.

- Ian Blake, Gadiel Seroussi & Nigel Smart (editors) (2005).
*Advances in Elliptic Curves in Cryptography*. Number 317 in London Mathematical Society Lecture Note Series. Cambridge University Press. ISBN 0-521-60415-X.

This extends Blake et al. (1999) in many directions and covers important recent results.

- Alfred Menezes (1993).
*Elliptic curve public key cryptosystems*. Kluwer Academic Publishers, Boston MA.

This book is a reference of relevant definitions and results. It provides only few proofs.

- Joseph H. Silverman (1986).
*The Arithmetic of Elliptic Curves*. Number 106 in Graduate Texts in Mathematics. Springer-Verlag, New York. ISBN 0-387-96203-4, 3-540-96203-4.

This is

*the*bible. Any detail that you could not find elsewhere, here there’s the way to it. However, this is the deepest and most mathematical of all books on this list and sometimes requires to look up other sources.

- BSI (2009).
*Technical Guideline TR-03111 Elliptic Curve Cryptography*. PDF. - Ian Connell (1999). Elliptic Curve Handbook.
- Henry Cohen, Gerhard Frey, Roberto Avanzi, Chritophe Doche, Tanja Lange, Kim Nguyen & Frederik Vercauteren (2005).
*Handbook of Elliptic and Hyperelliptic Curve Cryptography*.

This is

*the*reference. - Alfred J. Menezes, Yi-Hong Wu & Robert J. Zuccherato (1996). An elementary introduction to hyperelliptic curves. Technical Report CORR 96-19, Department of C&O, University of Waterloo, Ontario, Canada.

- http://www.cacr.math.uwaterloo.ca/conferences/2006/ecc2006/slides.html
- Wikipedia (2007). Elliptic Curve Cryptography.
- ECC 1997-2009 including slides from talks.
- Franz Lemmermeyer (2003). List of various lecture notes on elliptic curves.

## Mailinglist

We will put each member on the mailing list

## Allocation

4+2 SWS, 8 credits. Optionally, 3+2 SWS, 6 credits.

Successful completion of the course yields 8 credit points. For students who only want 6 credit points, a breakpoint at about 3/4 of the teaching time will be defined, and only the course material up to that point will be relevant for their exams and grades.

- Media Informatics: Computer and Communication Technology.
- Recommendation for University of Bonn - Computer Science: A or A1, respectively.