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

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 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)

 What Is Hasse's Theorem

 Finite Elliptic Curve Group, Eq(a,b), q = p^n

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB