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

Last update: 2019.

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)