Elliptic Curve Point Addition Example

This section provides algebraic calculation example of adding two distinct points on an elliptic curve.

Now we algebraic formulas to calculate the addition operation on elliptic curves. Let's try them with some examples.

The first example is adding 2 distinct points together, 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 two given points:
   P = (xP, yP) = (1,2)
   Q = (xQ, yQ) = (3,4)
  
Find the sum of P and Q: 
   R = P + Q = (xR, yR)
   
From equation (10):
       yP - yQ 
   m = ---------                (10)
       xP - xQ
       
We get: 
   m = -2/-2 = 1

From equations (8) and (9):
   xR = m2 - xP - xQ              (8)
   yR = m(xP - xR) - yP           (9)   
   
We get: 
   xR = 1*1 - 1 - 3 = -3
   yR = 1*(1 + 3) - 2 = 2   

So:
   R = (-3,2)

Here is how we can verify the result:

Point, -R=(-3,-2), must be on the elliptic curve, which can be verified as:
   y2 = x3 - 7x + 10
Or:
   (-2)*(-2) = (-3)*(-3)*(-3) - 7*(-3) + 10
   4 = -27 + 21 + 10 
   4 = 4 

Point, -R=(-3,-2), must be on the straight line passing through P and Q, 
which can be verified as:
   y = m(x - xP) + yP
Or:
   -2 = m(-3 - 1) + 2
   -2 = 1*(-4) + 2
   -2 = -2

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