The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 12

2.2 The Group Law

As we saw in Chapter 1, we could start with two points, or even one point,
on an elliptic curve, and produce another point. We now examine this process
in more detail.

P '

P 2

P 1

P 3

Figure 2.2
Adding Points on an Elliptic Curve

Start with two points

P 1 =( x 1 ,y 1 ) , P 2 =( x 2 ,y 2 )
on an elliptic curve E given by the equation y 2 = x 3 + Ax + B . Define a new
point P 3 as follows. Draw the line L through P 1 and P 2 . We'll see below that
L intersects E in a third point P 3 . Reflect P 3 across the x -axis (i.e., change
the sign of the y -coordinate) to obtain P 3 . We define

P 1 + P 2 = P 3 .
Examples below will show that this is not the same as adding coordinates of
the points. It might be better to denote this operation by P 1 + E P 2 , but we
opt for the simpler notation since we will never be adding points by adding
coordinates.
Assume first that P 1 = P 2 and that neither point is ∞ . Draw the line L
through P 1 and P 2 .Itsslopeis

y 2 − y 1

m =

x 1 .

x 2 −