Elliptic Curve Groups

  1. (15 minutes) Presentation - Perimeters of right triangles? Patterns in congruent numbers modulo $ 8$ ?
  2. (30 minutes) Definition of a group.

    Definition 4.1   An abelian group is a set $ X$ equipped with a binary operation $ +$ and an element $ 0\in X$ such that for all $ a,b,c \in X$ ,
    1. (closure) $ a+b \in X$ ,
    2. (identity) $ 0 + a = a + 0 = a$ ,
    3. (associativity) $ a + (b + c) = (a + b) + c$ ,
    4. (inverses) for every $ a$ in $ X$ there is $ d$ such that $ a+d = 0$ ,
    5. (commutativity) $ a + b = b + a$ .

    Examples:

    1. The integers $ \mathbb{Z}= \{0,-1,1,-2,2,-3,3,\ldots\}$ under addition.
    2. The rational numbers $ \mathbb{Q}$ under addition.
    3. The integers $ \{0,1,\ldots, n-1\}$ under addition modulo $ n$ .
    4. Let $ p$ be a prime. The integers $ \{1,\ldots, p-1\}$ under multiplication modulo $ p$ . This is called $ \mathbb{F}_p^*$ .

  3. (15 minutes) Experiment with some abelian groups in SAGE .

  4. (10 minutes) Break.

  5. (20 minutes) Definition of elliptic curve groups.

    Definition 4.2   Fix integers $ a$ and $ b$ . Let $ E(\mathbb{Q})$ be the set of solutions to $ y^2=x^3+ax+b$ along with one ``extra point'' which we call $ \mathcal{O}$ which is the additive 0 element. This is an abelian group (note: the associative law takes a lot of work to prove!).

  6. (30 minutes) Participants: Graph elliptic curves. Then derive an algebraic formula (by hand) for the group operation.

  7. (15 minutes) Elliptic curves modulo $ p$ . Fix integers $ a$ and $ b$ and a prime $ p$ . Let $ E(\mathbb{F}_p)$ be the set of solutions to $ y^2 \equiv x^3 + a x + b\pmod{p}$ with $ 0\leq x < p$ and $ 0\leq y < p$ along with a formal extra point $ \mathcal{O}$ . This group is central in both cryptography (in making and cracking cryptosystems) and the Birch and Swinnerton-Dyer conjecture! I will explain how in both cases next week.

  8. (15 minutes) Participants: Graph and compute with some elliptic curves modulo $ p$ .



Subsections
William Stein 2006-07-07