# Elliptic Curves: Theory and Practice

## An Informal View

The aim of this course is two-folded: understanding elliptic curves as a slice of algebraic geometry and refreshing our knowledge of some algorithms used in Cryptography and computer science in general.

The first part of the course is targeted to the math: We introduce algebraic varieties in both the projective and affine fashion and specialize to elliptic curves, seen as smooth curves of genus one. We then define a group structure on those and (if there is time) show that elliptic curves over the complex numbers are "the same as" complex tori. Later we analyze some aspects of elliptic curves (isogenies, Weil pairing, endomorphism ring).

The second part of the course is devoted to a description of some algorithms that make use of elliptic curves. We start by learning how to add points on an elliptic curve, then describe Lenstra's factorization algorithm and Shoof's algorithm to count points of elliptic curves over finite fields. Later, we have a glance at elliptic curve cryptography, pairing based cryptography and the computation of Weil pairing.

## The List

The books (suggestions on books are welcome!) I would like to follow are:

• The Arithmetic of Elliptic Curves, by Joseph H. Silverman.
• Prime Numbers, by Richard Crandall and Carl Pomerance.
• Cryptography - an Introduction, by Nigel Smart.

I was thinking of covering the following topics in at least five meetings - presenting for the first two (or more) myself.

### First Part

To be split into 2-3-? meetings.
• Affine and projective varieties and maps between them
• Divisors
• Weierstrass equations
• Elliptic curves (and complex tori?)
• Isogenies
• Weil pairing
• The endomorphism ring
• Number of rational points

### Second Part

To be split into 3-? meetings.
• Lenstra's EC factorization algorithm
• Counting points of elliptic curves over finite fields (Schoof's algorithm)
• Elliptic curve cryptography
• Elliptic curve discrete logarithm problem
• Pairing based cryptography
• Computing the Weil pairing

## Schedule

### 1st Meeting: "Finding the right amount of abstraction" - November 23rd

Topics:
• Definition of an EC (toy version vs. abstract version)
• Affine space
• Affine varieties
• Projective space
• Projective varieties
• Crandall-Pomerance: ch7 until Cassels' Thm (excluded).
• Silverman: ch1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 3.1.

### 2nd Meeting: "How did that group come out?" - December 2nd

Topics:
• Maps between varieties
• Examples
• Smoothness
• Isogenies
• Divisors
• EC over the complex?
• Complex tori?
• Weierstrass P-function?
• Group structure?
• Endomorphism Ring?
• Weil Pairing?
• Silverman: ch3.4.

### 3rd Meeting: "What else can we play with?" - December 7th

Topics:
• More on isogenies: degree
• Degree as expansion of the fundamental domain (EC over the complex)
• ECs over finite fields
• Number of points - a loose bound
• An inefficient algorithm to count points on EC/finite field
• The group law algorithm (easy version)
• An example over the rationals
• Mordell-Weil theorem (as a remark - Rank)
• Singular Weierstrass equations, group?