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

Every Element Is Subgroup Generator

This section describes the fact that every element in an finite Abelian group is a subgroup generator. The order of the generated subgroup is the same as the order of the element.

Another nice fact of an finite Abelian Group is that every element is a subgroup generator, which can be used to generate a subgroup.

To prove this fact, let's express it as:

Given an finite group G and any element P, the following element set is a subgroup generated by P: S = {P, 2P, 3P, ..., (n-1)P, nP=0} where n is the order of the element P.

Now we can verify if subgroup S satisfies the 5 rules of Abelian Group or not.

1. Closure:

If I = iP and J = jP are in S, Then K = I + J = iP + jP = (i+j)P is in S Proof: If i+j <= n, K = (i+j)P is in S by definition If i+j > n and i+j < 2n, we have: k = n - (i+j), where k > 0 and k < n i+j = n+k K = (i+j)P = (n+k)P = nP + kP = kP, since nP = 0 K = kP must be in S, because k < n

2. Commutativity:

If I = iP and J = jP are in S, Then I + J = J + I Proof: I and J are elements in the parent group, so: I + J = J + I, Commutativity of the parent group

3. Associativity:

If I = iP, J = jP and K = kP are in S, Then (I + J) + K = I + (J + K) Proof: I, J and K are elements in the parent group, so: (I + J) + K = I + (J + K), Associativity of the parent group

4. Identity Element:

If I = iP is in S, Then iP = nP = iP Proof: iP is an element in the parent group and 0 is the identity element, so: iP + nP = iP + 0 = iP, where i = 1, 2, ..., n-1

5. Symmetry:

If I = iP is in S, there is j exists, such that: iP + jP = 0 Proof: i <= n by definition If i = n, then j = n, and iP + jP = nP + nP = 0 + 0 = 0 If i < n, then j = n - i, iP + jP = iP + (n-i)P = nP = 0

Note that the order of the subgroup (group size) is the same as the order of the element which generated the subgroup.

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

What Is Subgroup in Abelian Group

What Is Subgroup Generator in Abelian Group

►Every Element Is Subgroup Generator

Order of Subgroup and Lagrange Theorem

Element Generated Subgroup Is Cyclic

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)