# Coursebooks

## Algebraic curves and cryptography

English

#### Remarque

Cours donnés en alternance tous les deux ans (pas donné en 2019-20)

#### Summary

The goal of this course is to introduce basic notions from public-key cryptography based on algebraic curves over finite fields. We will introduce basic cryptographic schemes as well as discuss in-depth the discrete logarithm problem for elliptic and Jacobians of higher genus curves.

#### Content

Topics may include, but are not limited to:
• Introduction to algebraic curves
• Elliptic and hyperelliptic curves
• Jacobians of algebraic curves
• Cantor arithmetic
• Elliptic curve discrete logarithm problem
• Index calculus methods for Jacobians
• Pairing-based cryptography

#### Keywords

algebraic curves over finite fields, public key cryptography, discrete logarithms, pairing-based cryptography

#### Learning Prerequisites

##### Required courses

Abstract Algebra required (groups theory, rings, fields, field extensions, finite fields)

##### Recommended courses

• Math 317 (Galois theory)
• Math 489 (Number Theory in Cryptography)
• COM-401 (Security and Cryptography)

#### Teaching methods

Weekly lectures, problem sets and programming assignments.

#### Assessment methods

written exam

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

#### Resources

##### Bibliography

• P. Griffiths, Introduction to Algebraic Curves
• I. Blake, G. Seroussi, and N. Smart, Elliptic Curves in Cryptography
• I. Blake, G. Seroussi, N. Smart, Advances in Elliptic Curve Cryptography

### In the programs

• Mathematics - master program, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Communication Systems - master program, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Communication Systems - master program, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science - Cybersecurity, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science - Cybersecurity, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic curves and cryptography
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
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• Lecture in French
• Lecture in English
• Lecture in German