EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, by Dr. Herong Yang
Elliptic Curve Geometric Properties
This section describes two geometric properties of an elliptic curve: horizontal symmetry and 3 intersections or less with straight lines.
Elliptic curves have two important geometric properties:
1. An elliptic curve is symmetrical about the x-axis. If point P = (x,y) is on an elliptic curve, then -P = (x,-y) is also on the same elliptic curve. We can also say that an elliptic curve is horizontally symmetric.
Here is an example of two symmetrical points P and -P on an elliptic curve (source: slideplayer.com):
2. An elliptic curve will intersect with any straight line in at most 3 points. This property will be used to help defining an addition operation on an elliptic curve.
Here is an example of an elliptic curve intersecting with a straight line in 3 points (source: sysax.com):
Table of Contents
►Geometric Introduction to Elliptic Curves
►Elliptic Curve Geometric Properties
Addition Operation on an Elliptic Curve
Prove of Elliptic Curve Addition Operation
Same Point Addition on an Elliptic Curve
Infinity Point on an Elliptic Curve
Negation Operation on an Elliptic Curve
Subtraction Operation on an Elliptic Curve
Identity Element on an Elliptic Curve
Commutativity of Elliptic Curve Operations
Associativity of Elliptic Curve Operations
Elliptic Curve Operation Summary
Algebraic Introduction to Elliptic Curves
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)