**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.03, by 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.

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)