Group: A tuple of a set and an operation

Definition

Group is tuple of a set & an operation $o$ (Every time students hover cursor over that group, we show what 2 things it is defined on) like $\mathbb{G} = (\mathbb{R}, +)$,

Why Matters?

Why Concept of Group Matters in Cryptography

Group makes sure that performing operation on numbers follows specific rules, making it reliable for cryptographic system. Through this Group framework, a certain operation like "modulo" can help secure our cryptographic system. (More on this in Modulo Section)

Properties

Group $\mathbb{G}$ must have these 4 following properties: (We need to make sure this "Group" is not just word in english but represent Mathematical structure --> can do by allowing us to hover cursor and it clearly shows it's a group, unlike when we hover cursor over word "group" and nothing happens.)

1. Closure Property

Definition

If you take any two numbers from the set and apply the operation, you'll always get a number that's still within the set.

Math Expression

Given $(\mathbb{G}, o)$, For all $a,b \in \mathbb{G}$, $(a\ o\ b) \in \mathbb{G}$

We start with definition as "intuitive word", then showing "Math Expression" where we write it more succinctly using Math to onboard people to learn what those notations mean.

Notation

"For All" can be written as $\forall$. Example: For all a,b $\in$ G, can be written as $\forall$ a,b $\in$ G

(Here every time this notation is shown later, we can let students click and it hop back here where its definition is explained, or can just allow students to hover cursor above this notation and show this definition, so they dont need to scroll back and forth.)

2. Associativity Property

Definition

No matter how you place the parentheses when combining numbers, you'll get the same outcome.

Math Expression

Given $(\mathbb{G}, o)$, $\forall$ $a,b,c \in \mathbb{G}$, $(a\ o\ b)\ o\ c = a\ o\ (b\ o\ c)$

3. Identity Property

Definition

There's a special number in the set that doesn't change other numbers when used in the operation like adding 0 or multiplying by 1 -- you get the original number back.

Math Expression

There exists $e \in \mathbb{G}$ such that $\forall a \in \mathbb{G},\ e\ o\ a = a\ o\ e = a$. We call this 'e' as 'Identity Element'

Notation

"There exists" can be written as $\exists$. Example: There exists $e \in \mathbb{G}$, can be written as $\exists e \in \mathbb{G}$

4. Inverse Property

Definition

For every element in the set, there exists another element in the set that can be combined with it to produce the identity element, like adding a number and its negative to get 0, or multiplying a number by its reciprocal to get 1.

Math Expression

$\forall a \in \mathbb{G} , \exists a^{-1} \in \mathbb{G}$ such that $a\ o\ a^{-1} = a^{-1}\ o\ a = e$. We call this $a^{-1}$ as an inverse of $a$