Generators and Cyclic Subgroups
This chapter provides introduction on generating subgroups from elements in finite Abelian groups; definitions and examples of subgroup generators, subgroup order, cyclic subgroups, Lagrange-Theorem.
What Is Subgroup in Abelian Group
What Is Subgroup Generator in Abelian Group
What Is Order of Element
Every Element Is Subgroup Generator
Order of Subgroup and Lagrange Theorem
What Is Cyclic Group
Element Generated Subgroup Is Cyclic
- A subgroup in an Abelian Group is a subset of the Abelian Group that
itself is an Abelian Group.
- A subgroup generator is an element in an finite Abelian Group that can be
used to generate a subgroup using a series of scalar multiplication operations
in additive notation.
- The order of an element P in a finite Abelian Group is the least
positive integer n, such that the scalar multiplication of n and P is 0,
where 0 is the identity element in additive notation.
- Every element in a finite Abelian group is a subgroup generator,
which can be used to generate an subgroup.
- Lagrange Theorem states that the order of any subgroup in an
finite Abelian group divides the order of the parent group.
- A Cyclic Subgroup is a finite Abelian group that can be generated by
a single element using the scalar multiplication operation in additive
notation (or exponentiation operation in multiplicative notation).
- All subgroups generated from elements in finite Abelian groups are cyclic groups.
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)
►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
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