Galois Field Multiplication

For a galois field of $GF(2^m)$, we can represent bits as a Polynomial and perform Polynomial Multiplication on them, divide by a constant polynomial and get the remainder.

Example

For bits:

• $000$ can be represented as polynomial $0$
• $001$ can be represented as polynomial $1$
• $010$ can be represented as polynomial $x$
• $011$ can be represented as polynomial $x+1$
• $100$ can be represented as polynomial $x^2$
• $101$ can be represented as polynomial $x^2+1$
• $110$ can be represented as polynomial $x^2+x$
• $111$ can be represented as polynomial $x^2+x+1$ For multiplying $100\times 011$,
• $x^2\times (x+1)=x^3+x^2$
• This is $1100$