**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.00, by Dr. Herong Yang

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: y^{2}= x^{3}+ ax + b, where (a=-7 and b=10) Or: y^{2}= x^{3}- 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}= m^{2}- 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: y^{2}= x^{3}- 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

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)