EC Cryptography Tutorials - Herong's Tutorial Examples

∟Elliptic Curve Subgroups

This chapter provides notes on subgroup generation from reduced elliptic curve groups, Ep(a,b). Python programs are provided to perform point addition, scalar multiplication, and subgroup generation.

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Subgroups of Reduced Elliptic Curve Groups

Python Program for Point Addition

Point Addition Extended with Identity

Python Program for Scalar Multiplication

Python Program for Generating Subgroup

Subgroup Example - G(15,13) of E17(0,7)

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Takeaways:

• Subgroup groups can be generated with a base point from a reduced elliptic curve group, since it is a finite Abelian group.
• It is not so difficult to write Python programs to to perform point addition, scalar multiplication, and subgroup generation..

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

►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)

 EC Cryptography in Java

 Standard Elliptic Curves

 Terminology

 References

 Full Version in PDF/EPUB

Elliptic Curve Subgroups - Updated in 2022, by Dr. Herong Yang