Curves defined by equations

Warm-up Exercise. Sketch the graphs of the curves defined by the following equations.

1. \(x^2+y^2=16\)
2. \(x^2+4y^2=16\)
3. \(x^2+y^2=x\)

Elliptic Curves

An elliptic curve is the set of solutions to an equation of the form \[ y^2 = x^3 + bx + c \] over some field \(F\).

Applications

Experiment I

See handout.

Addition: Geometric definition

Suppose \(A\) and \(B\) are points on an elliptic curve.

1. Draw the line through \(A\) and \(B\).
2. This line intersects the curve in the point \(P\).
3. Reflect \(P\) over the \(x\)-axis to obtain the point \(A + B\).

(Board)

Questions:

• Is this definition of addition commutative?
• What if \(A\) and \(B\) are the same point?

Experiment II

Group of points on an elliptic curve

Computational example

Curve: \(y^2 = x^3 + 73\)

1. Add \((2,9) + (3,10)\).
2. Add \((-4,-3) + (-4,-3)\).

Upshot

• Once you get the slope \(m\) of the line, it is easy to find \(x_3\), the \(x\)-coordinate of the intersection, using the “sum of the roots” trick.
• Once you get \(x_3\), it is easy to use the equation of the line to get \(y_3\).

Addition: Algebraic formulas

Addition Law. If \(P\) is any point on an elliptic curve, then \(P + \infty = P = \infty + P\). Let \(P_1 = (x_1,y_1)\) and \(P_2 = (x_2,y_2)\) be points on the elliptic curve \(y^2 = x^3 + bx +c\).

If \(P_1 = P_2\) and \(y_1 = 0\), then \(P_1+P_2 = \infty\).

Otherwise, \(P_1 + P_2 = P_3 = (x_3,y_3)\), where \[ \begin{aligned} x_3 &= m^2 -x_1 -x_2 \\ y_3 &= m(x_1-x_3)-y_1 \end{aligned} \] and \[ m = \left\{\begin{array}{cl} (y_2-y_1)/(x_2-x_1) & \mbox{if } P_1 \neq P_2 \\ (3x_1^2+b)/(2y_1) & \mbox{if } P_1 = P_2 \end{array} \right. \]