# Math Help - how to do arithmetic operations in galois field

1. ## how to do arithmetic operations in galois field

is primitive in GF(2)[x].
Let be a root of .

α2 means 0 1 0 0
α5 means 0 1 1 1
- ----------------
when we add the both 1 0 1 1

but in galois field answer is α3 means that 0 0 1 1 from the table

Multiplication:
in multiplication when we multiply two 4 bit numbers we got 8 bit number
in galois field we got 4 bit number only

Because $\alpha$ is a root of $x^3+x+1$.
Therefore $\alpha^3 + \alpha + 1 = 0 \implies \alpha^3 = - \alpha - 1$.
But this field has charachteristic two i.e. $x=-x$.
Thus, $\alpha^3 = -\alpha - 1 = \alpha + 1$.