18.783 Elliptic Curves
Lecture #13
13
Spring 2017
03/22/2017
Endomorphism algebras
The key to improving the eciency of elliptic curve primality proving (and many other
algorithms) is the ability to di
18.783 Elliptic Curves
Lecture #11
11
Spring 2017
03/15/2017
Index calculus, smooth numbers, and factoring integers
Having explored generic algorithms for the discrete logarithm problem in some detail
18.783 Elliptic Curves
Lecture #16
16
Spring 2017
04/10/2017
Elliptic curves over C (part 2)
Last time we showed that every lattice L C gives rise to an elliptic curve
EL : y 2 = 4x3 g2 (L)x g3 (L),
w
18.783 Elliptic Curves
Lecture #7
7
7.1
Spring 2017
03/01/2017
Torsion subgroups and endomorphism rings
The n-torsion subgroup E[n]
Having determined the degree and separability of the multiplication-
18.783 Elliptic Curves
Lecture #3
3
Spring 2017
02/15/2017
Finite elds and integer arithmetic
In order to perform explicit computations with elliptic curves over nite elds, we rst need
to understand a
18.783 Elliptic Curves
Lecture 1
Andrew Sutherland
February 8, 2017
1
What is an elliptic curve?
The equation
x2
a2
+
y2
b2
= 1 denes an ellipse.
An ellipse, like all conic sections, is a curve of gen
18.783 Elliptic Curves
Lecture #5
5
Spring 2017
02/22/2017
Isogenies
In almost every branch of mathematics, when considering a category of mathematical objects with a particular structure, the maps be
18.783 Elliptic Curves
Lecture #12
12
Spring 2017
03/20/2017
Primality proving
In this lecture, we consider the following problem: given a positive integer N , how can we
eciently determine whether N
18.783 Elliptic Curves
Lecture #4
4
Spring 2017
02/21/2017
Finite eld arithmetic
We saw in Lecture 3 how to eciently multiply integers, and, using Kronecker substitution,
how to eciently multiply poly
18.783 Elliptic Curves
Lecture #25
Spring 2017
05/15/2017
Modular forms and L-functions
25
As we will prove in the next lecture, Fermats Last Theorem is a corollary of the following
theorem for ellipt
18.783 Elliptic Curves
Lecture #22
22
Spring 2017
05/03/2017
Ring class elds and the CM method
Let O be an imaginary quadratic order of discriminant D, and let EllO (C) := cfw_j(E) C :
End(E) = C. In
18.783 Elliptic Curves
Lecture #14
14
Spring 2017
04/03/2017
Ordinary and supersingular elliptic curves
Let E/k be an elliptic curve over a eld of positive characteristic p. In Lecture 7 we proved
tha
18.783 Elliptic Curves
Lecture #9
9
Spring 2017
03/08/2017
Schoofs algorithm
In the early 1980s, Ren Schoof [3, 4] introduced the rst polynomial-time algorithm to
compute #E(Fq ). Extensions of Schoof
18.783 Elliptic Curves
Lecture #17
17
Spring 2017
04/12/2017
Complex multiplication
Over the course of the last two lectures we established a one-to-one correspondence between
lattices L C (up to home
18.783 Elliptic Curves
Lecture #20
20
Spring 2017
04/26/2017
The modular equation
In the previous lecture we defined modular curves as quotients of the extended upper half
plane under the
subgroup (a
18.783 Elliptic Curves
Lecture #24
24
Spring 2017
05/10/2017
Divisors and the Weil pairing
In this lecture we address a completely new topic, the Weil Pairing, which has many practical
and theoretical
18.783 Elliptic Curves
Lecture #26
26
Spring 2017
05/17/2017
Fermats Last Theorem
In this nal lecture we give an overview of the proof of Fermats Last Theorem. Our goal
is to explain exactly what Andr
18.783 Elliptic Curves
Lecture #2
2
Spring 2017
02/15/2017
Elliptic curves as abelian groups
In Lecture 1 we dened an elliptic curve as a smooth projective curve of genus 1 with a
distinguished ration
18.783 Elliptic Curves
Lecture #18
18
Spring 2017
04/19/2017
The CM torsor
Over the course of the last three lectures we have established an equivalence of categories
between complex tori C/L and elli
18.783 Elliptic Curves
Lecture #15
15
Spring 2017
04/05/2017
Elliptic curves over C (part I)
We now consider elliptic curves over the complex numbers. Our main tool will be the
correspondence between
18.783 Elliptic Curves
Lecture #6
6
Spring 2017
02/27/2017
Isogeny kernels and division polynomials
In this lecture we continue our study of isogenies of elliptic curves. Recall that an isogeny
is a s
18.783 Elliptic Curves
Lecture #21
21
Spring 2017
05/01/2017
The Hilbert class polynomial
In the previous lecture we proved that the eld of modular functions for 0 (N ) is generated
by the functions j
18.783 Elliptic Curves
Lecture #10
10
Spring 2017
03/13/2017
Generic algorithms for the discrete logarithm problem
We now consider generic algorithms for the discrete logarithm problem in the standard
18.783 Elliptic Curves
Lecture #19
19
Spring 2017
04/24/2017
Riemann surfaces and modular curves
Let O be an order in an imaginary quadratic eld and let cl(O) be its ideal class group
(proper O-ideals
18.783 Elliptic Curves
Lecture #23
23
Spring 2017
05/08/2017
Isogeny volcanoes
We now shift our focus from elliptic curves over C to elliptic curves other elds, and to
nite elds in particular. As note
18.783 Elliptic Curves
Lecture #8
8
Spring 2017
03/06/2017
Point counting
8.1
Hasses Theorem
We are now ready to prove Hasses theorem.
Theorem 8.1 (Hasse). Let E/Fq be an elliptic curve over a finite
INSTITUT NATIONAL DES SCIENCES A PPLIQUEES DE TOULOUSE
DpartementdeSciencesetTechnologiesPourl'Ingnieur
3me anneIngnieriedelaConstruction
GOTECHNIQUE1
CoursChapitres1
JacquesLrau
MatredeConfrences
Ann
Matriaux.
_
Ren Motro
Universit Montpellier II
Stabilit des ouvrages/ Matriaux. Ren Motro. 2005
Introduction
Galile : traction sur des fils mtalliques
Hooke : lasticit linaire
Young : module de dfo