**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, by Dr. Herong Yang

Elliptic Curves with Singularities

This section describes elliptic curves with singularities where curves are not smooth.

If we review the elliptic curve equation again, we will see that for certain combination of coefficients a and b, the elliptic curve will a singularity point where the curve is not smooth.

y^{2}= x^{3}+ ax + b

Here are two examples of elliptic curves with singularities (source: andrea.corbellini.name):

To exclude curves with singularities, we need to add an extra condition on coefficients a and b:

y^{2}= x^{3}+ ax + b 4a^{3}+ 27b^{2}!= 0

Last update: 2019.

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)