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What is a Ring?

A Ring is a set of integers ($\Bbb Z$) with a size ($m$).

\[\Bbb Z_m = \{ 0,1,2,\dots m-1 \}\]

Rings have associated Operators with special rules.

• Addition: ‘+’
• Multiplication: ‘*’

For all $a,b\in \Bbb Z_m$:

• $a + b \equiv c \bmod m$
• $a * b \equiv d \bmod m$

Properties of Rings

Closure

Note that $c,d \in \Bbb Z_m$

Associative

• $(a+b)+c=a+(b+c)$
• $(a\cdot b)\cdot c=a\cdot (b\cdot c)$

Identity

• $a+0=a\mod m$
• $a\cdot 1=a\mod m$

Inverse

• $a+(-a)=0\mod m$
• $a\cdot a^{-1}=1\mod m$

Note that $a^{-1}$ only exists when $\gcd (a, m)=1$

You can find $a^{-1}$ with the extended euclidean algorithm