EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, by Dr. Herong Yang
Converting Elliptic Curve Groups
This section describes steps on how to convert real number elliptic curve groups to cyclic subgroups of integer elliptic curve groups.
In previous chapters, we have learned how to construct Abelian groups on elliptic curves, and build trapdoor function based on DLP (Discrete Logarithm Problem) on elliptic curve groups. We have also learned what are finite fields and cyclic subgroups of finite groups.
Now we need to learn how to construct cyclic subgroups on elliptic curves so that their trapdoor functions can be used in cryptography technologies. We will do this in 3 steps:
Step 1: Real number elliptic curve groups - Construct infinite Abelian groups with all real number points on an elliptic curve with the rule of chord operation. Those points form a continuous curve in 2-dimensional real number space.
Real number elliptic curve groups has been discussed already in previous chapters.
Step 2: Finite elliptic curve groups - Construct finite Abelian groups with integer points on an elliptic curve reduced by modular arithmetic with the rule of chord operation. Those points are bounded in an rectangular area in 2-dimensional integer space.
We will discuss finite elliptic curve groups in this chapter.
Step 3: Cyclic elliptic curve groups - Construct cyclic Abelian groups by selecting base points from Finite elliptic curve groups and generate cyclic subgroups.
We will discuss cyclic elliptic curve groups in the next chapter.
Table of Contents
Geometric Introduction to Elliptic Curves
Algebraic Introduction to Elliptic Curves
Abelian Group and Elliptic Curves
Discrete Logarithm Problem (DLP)
Generators and Cyclic Subgroups
►Reduced Elliptic Curve Groups
►Converting Elliptic Curve Groups
Elliptic Curves in Integer Space
Python Program for Integer Elliptic Curves
Elliptic Curves Reduced by Modular Arithmetic
Python Program for Reduced Elliptic Curves
Point Pattern of Reduced Elliptic Curves
Integer Points of First Region as Element Set
Reduced Point Additive Operation
Modular Arithmetic Reduction on Rational Numbers
Reduced Point Additive Operation Improved
What Is Reduced Elliptic Curve Group
Reduced Elliptic Curve Group - E23(1,4)
Reduced Elliptic Curve Group - E97(-1,1)
Reduced Elliptic Curve Group - E127(-1,3)
Reduced Elliptic Curve Group - E1931(443,1045)
Finite Elliptic Curve Group, Eq(a,b), q = p^n
tinyec - Python Library for ECC
ECDH (Elliptic Curve Diffie-Hellman) Key Exchange
ECDSA (Elliptic Curve Digital Signature Algorithm)