Elliptic Curve Point Doubling Example

EC Cryptography Tutorials - Herong's Tutorial Examples

∟Algebraic Introduction to Elliptic Curves

∟Elliptic Curve Point Doubling Example

This section provides algebraic calculation example of point doubling, adding a point to itself, on an elliptic curve.

The second example is doubling a single point, also taken from "Elliptic Curve Cryptography: a gentle introduction" by Andrea Corbellini at andrea.corbellini.name/2015/05/17 /elliptic-curve-cryptography-a-gentle-introduction/:

For the elliptic curve given below:
   y² = x³ + ax + b, where (a=-7 and b=10)
Or:
   y² = x³ - 7x + 10

And a given point:
   P = (x_P, y_P) = (1,2)

Find the sum of P and P or 2P:
   R = 2P = (x_R, y_R)

From equation (6):
       3(x_P)² + a
   m = ---------                 (6)
         2(y_P)

We get:
   m = (3*1*1-7)/4 = -4/4 = -1

From equations (4) and (5):
   x_R = m² - 2x_P                  (4)
   y_R = m(x_P - x_R) - y_P           (5)

We get:
   x_R = (-1)*(-1) - 2*1 = -1
   y_R = (-1)*(1 + 1) - 2 = -4

So:
   R = (-1,-4)

Here is how we can verify the result:

Point, -R=(-1,4), must be on the elliptic curve, which can be verified as:
   y² = x³ - 7x + 10
Or:
   4*4 = (-1)*(-1)*(-1) - 7*(-1) + 10
   16 = -1 + 7 + 10
   16 = 16

Point, -R=(-1,4), must be on the straight line passing through P,
and tangent to the curve, which can be verified as:
   y = m(x - x_P) + y_P
Or:
   4 = m(-1 - 1) + 2
   4 = -1*(-2) + 2
   4 = 4