**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.02, by Dr. Herong Yang

Elliptic Curve Point Addition Example

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

Now we have 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: y^{2}= x^{3}+ ax + b, where (a=-7 and b=10) Or: y^{2}= x^{3}- 7x + 10 And two given points: P = (x_{P}, y_{P}) = (1,2) Q = (x_{Q}, y_{Q}) = (3,4) Find the sum of P and Q: R = P + Q = (x_{R}, y_{R}) From equation (10): y_{P}- y_{Q}m = --------- (10) x_{P}- x_{Q}We get: m = -2/-2 = 1 From equations (8) and (9): x_{R}= m^{2}- x_{P}- x_{Q}(8) y_{R}= m(x_{P}- x_{R}) - y_{P}(9) We get: x_{R}= 1*1 - 1 - 3 = -3 y_{R}= 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: y^{2}= x^{3}- 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 - x_{P}) + y_{P}Or: -2 = m(-3 - 1) + 2 -2 = 1*(-4) + 2 -2 = -2

Table of Contents

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)

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)