I am practicing some cryptography problems and I am having problems with one involving Galois Fields and irreducible polynomials. Here is the problem:
Using the irreducible polynomial $f(x) = x^5 + x^4 + x^3 + x^2 + 1$ to
a) generate the elements of the field $GF(2^5)$
b) based on the results of a, calculate the following in $GF(2^5)$:
1b. $(x^4 - x + 1)^{-1}$
2b. $(x^3 - x + 1) * (x^4 + x^2 + 1)$
3b. $(x^4 - x^3 + 1) / (x^2 + x + 1)$
I am not exactly sure how to even start this problem. If anyone can please help me, I would appreciate it very much! This is what I have:
$x^5 = x^4 + x^3 + x^2 + 1 \Rightarrow {F}={1111}$
$0 = 0 0 0 0 0$
$1 = 0 0 0 0 1$
$g = 0 0 0 1 0$
$g^2 = 0 0 1 0 0$
$g^3 = 0 1 0 0 0$
$g^4 = 1 0 0 0 0$
$g^5 = g^4 + g^3 + g^2 + 1 \rightarrow 1 1 1 0 1 $
$g^6 = 0 0 1 1 1$
$g^7 = 0 0 0 1 0$
$g^8 = 1 0 0 0 0$
...
I'm not exactly sure if I am even doing this correctly. Please help! Thanks!
It seems like you want your five-bit string $b_4b_3b_2b_1b_0$ to correspond to the (equivalence class of the) polynomial $b_4x^4+b_3x^3+b_2x^2+b_1x+b_0$. Therefore $0$ is indeed 00000, but $1$ is 00001 and $g$ is 00010. You have $g^2$ through $g^4$ right, but $g^5$ is 11101, not 1111.