## Composite field intro

From this paper:

There is only one finite field of characteristic 2 for a given degree, and both the binary and composite fields refer to the same field although their representation methods are different.”

$GF(2^{k})$
$B_{1}=\left \{ 1, \alpha , \alpha ^{2}, ... \alpha ^{k-1} \right \}$
$A \in GF\left ( 2^{k} \right )$
$A = \sum_{i=0}^{k-1} a_{i}.\alpha ^{i}$
where $a_{0}, a_{2}, ... a_{k-1} \in GF\left ( 2 \right )$

————–

$GF((2^n)^m)$
$B_2 = {1, \beta, \beta ^2, .... \beta ^{m-1}}$
$A \in GF((2^n)^m)$
$A=\sum_{i=0}^{m-1}a'_i.\beta ^i$
where $a'_0, a'_1, ...a'_m-1 \in GF(2^n)$

Examples:

$GF(2^8)$
$B_1 = {1, \alpha, \alpha^2, ..., \alpha^7}$
$A \in GF(2^8)$
$A = \sum_{i= 0}^{7}a_i.\alpha^i$
$a_0, a_1, ..., a_7 \in GF(2)$

————–

$GF((2^2)^4)$
$B_2 = {1, \beta, \beta^2, \beta^3}$
$A \in GF((2^2)^4)$
$A = \sum{i=0}^{2}a'_i.\beta^i$
$a'_0, a'_1, a'_2 \in GF(2^2)$

There are still I’d like to add here but don’t have time to type it in excel or draw it using omnigraffle. I’ll just keep it on my phd logbook. Now going to read some more papers. And start coding.