Algebraic Solution for Distinct Points

EC Cryptography Tutorials - Herong's Tutorial Examples

∟Algebraic Introduction to Elliptic Curves

∟Algebraic Solution for Distinct Points

This section provides an algebraic solution for calculating the addition operation of two distinct points on an elliptic curve.

Case 3: P and Q are two distinct points.

Let m be the slope of the straight line passing through P,
the line can be expressed as the following equation:
   y = m(x - x_P) + y_P

The other intersection, -R, of the line and the curve must satisfy
the line equation and the curve equation:
   -y_R = m (x_R  - x_P) + y_P        (1)
   (-y_R)² = (x_R)³ + a(x_R) + b     (2)

Note that -y_R in equation (2) can be replaced by the right
hand side of equation (1):
   (m(x_R - x_P) + y_P)² = (x_R)³ + a(x_R) + b

Regrouping terms will result a cubic equation for x_R:
   (x_R)³ - m²(x_R)² + (a - 2m(y_P) + 2m²(x_P))(x_R) + b - (m(x_P)-y_P)² = 0

We know that a cubic equation has 3 roots and their sum is the negation
of the coefficient of the second term based on Vieta's first formula,
see Vieta's formulas at wikipedia.org:
   x_P + x_Q + x_R = m²              (7)

Rearrange terms of equations (1) and (7) gives us the simplest way
to calculate, R:
   x_R = m² - x_P - x_Q              (8)
   y_R = m(x_P - x_R) - y_P           (9)

The slope of the line, m, can be determined by the other point, Q,
on the line:
       y_P - y_Q
   m = ---------                (10)
       x_P - x_Q

Now we can calculate R = P + Q with equations (8), (9) and (10) with any given distinct points, P and Q.

By the way, proof of Vieta's formulas can be found in "Cubic Equations" by Arkajyoti Banerjee and Ayush G Rai at https://brilliant.org/wiki/cubic-equations/