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 - xP) + yP

The other intersection, -R, of the line and the curve must satisfy 
the line equation and the curve equation:
   -yR = m (xR  - xP) + yP        (1)
   (-yR)2 = (xR)3 + a(xR) + b     (2)

Note that -yR in equation (2) can be replaced by the right
hand side of equation (1):
   (m(xR - xP) + yP)2 = (xR)3 + a(xR) + b 
   
Regrouping terms will result a cubic equation for xR:
   (xR)3 - m2(xR)2 + (a - 2m(yP) + 2m2(xP))(xR) + b - (m(xP)-yP)2 = 0

We know that a cubis 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:
   xP + xQ + xR = m2              (7)

Rearrange terms of equations (1) and (7) gives us the simpliest way 
to calculate, R:
   xR = m2 - xP - xQ              (8)
   yR = m(xP - xR) - yP           (9)   
   
The slope of the line, m, can be determined by the other point, Q, 
on the line:
       yP - yQ 
   m = ---------                (10)
       xP - xQ

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/

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

 Algebraic Description of Elliptic Curve Addition

 Algebraic Solution for Symmetrical Points

 Algebraic Solution for the Infinity Point

 Algebraic Solution for Point Doubling

Algebraic Solution for Distinct Points

 Elliptic Curves with Singularities

 Elliptic Curve Point Addition Example

 Elliptic Curve Point Doubling Example

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB