Lecture 2 Notes: The elliptic curve group law
Recall from Lecture 1 the de_ning property of the group law for an elliptic curve de_ned
by a Weierstrass equation y 2 = x3 + Ax + B:
Three points on a line sum to zero, which is the point at in_nity.
It is ea
Lecture 6 Notes: Division Polynomials
In this lecture we continue our study of isogenies and introduce the division polynomials.
Recall that an isogeny is a rational map that is also a group homomorphism. In the last
lecture we showed that every nonzero i
Lecture 3 Notes: Integer Arithmetics
To make explicit computations with elliptic curves over _nite _elds, we need to know how
to perform arithmetic operations in _nite _elds, and we would like to do so as e_ciently as
possible. In the applications we will
Lecture 7 Notes: The n-torsion subgroup
De_nition 7.1. Let G be an additive abelian group. The n-torsion subgroup G[n] is the
the kernel of the multiplication-by-n homomorphism [n], the set fg 2 G : ng = 0g.
We can now determine the structure of the n-tor
Lecture 5 Notes: Isogenies
We spent the last four lectures focusing how to e_ciently compute the group operation
for an elliptic curve E over a _nite _eld Fq . Let's take a moment to look at the bigger
picture. Given E=Fp there are several questions we mi
Lecture 10 Notes: The Discrete Logarithm Problem
In its most standard form, the discrete logarithm problem (DLP) is stated as follows:
Given 2 G and _ 2 hi, _nd the least positive integer x such that
x
= _.
In additive notation, we want x = _. In any case
Lecture 11 Notes: Index Calculus
We now give a lower bound for solving the discrete logarithm problem with a generic group
algorithm. We will show that if p is the largest prime divisor of N , then any generic
group algorithm for the discrete logarithm pr
Lecture 9 Notes: Schoof 's Algorithm
In the early 1980's, Ren_e Schoof introduced the _rst polynomial-time algorithm to compute #E(Fq ). Extensions of Schoof's algorithm remain the point-counting method of choice
when the characteristic of Fq is large (e.
Lecture 8 Notes: Point counting
We now consider the problem of determining the number of points on an elliptic curve E
over a _nite _eld Fq . The most na_ve approach one could take would be to evaluate
the curve equation y 2 = x3 + Ax + B for E at every p