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

Algebraic Solution for Symmetrical Points

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

Case 1: P and Q are symmetrical points - This case is very easy.

If P and Q are symmetrical points, we know that: y_{Q}= - y_{P}P + Q = P + (-P) = O (the infinity point or identity element) So: R = O = (∞, ∞) x_{R}= ∞ y_{R}= ∞

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)