GF
is small enough that you should construct an antilog table for it and save it for later reference rather than compute the polynomial form of 
or 
on the fly each time you need it. The
computer version of the antilog table is an array that stores the polynomial forms for 
in locations 
. For human use, the table
is constructed with two columns and looks something like this
$$\begin{array}{r|l}
\hline\\
i & \alpha^i \text{ equals}\\
\hline\\
0 & 00000001\\
1 & 00000010\\
2 & 00000100\\
3 & 00001000\\
4 & 00010000\\
5 & 00100000\\
6 & 01000000\\
7 & 10000000\\
8 & 00011101\\
9 & 00111010\\
10 & 01110100\\
11 & 11101000\\
12 & 11001101\\
\vdots & \vdots\quad \vdots \quad \vdots\\
254 & 10001110\\
\hline
\end{array}$$
The 
-th entry in the second column is the polynomial representation
of 
in abbreviated format. For example, 
is stated to be equal to 
which is shorthand for 
.
The entry for 
is obtained by shifting the entry
immediately above by one place to the left (inserting a 
on the
right) and if there is an 
term thus formed, removing it
and adding 
(i.e. XORing 
)
into the
rightmost 
bits. This process is easy to mechanize to produce
the antilog table by computer rather than by hand (which can be tedious
and mistake-prone).
GF is small enough that you should construct an antilog table for it and save it for later reference rather than compute the polynomial form of or on the fly each time you need it. The
computer version of the antilog table is an array that stores the polynomial forms for in locations . For human use, the table
is constructed with two columns and looks something like this
$$\begin{array}{r|l}
\hline\\
i & \alpha^i \text{ equals}\\
\hline\\
0 & 00000001\\
1 & 00000010\\
2 & 00000100\\
3 & 00001000\\
4 & 00010000\\
5 & 00100000\\
6 & 01000000\\
7 & 10000000\\
8 & 00011101\\
9 & 00111010\\
10 & 01110100\\
11 & 11101000\\
12 & 11001101\\
\vdots & \vdots\quad \vdots \quad \vdots\\
254 & 10001110\\
\hline
\end{array}$$
The -th entry in the second column is the polynomial representation
of in abbreviated format. For example, is stated to be equal to which is shorthand for .
The entry for is obtained by shifting the entry
immediately above by one place to the left (inserting a on the
right) and if there is an term thus formed, removing it
and adding (i.e. XORing )
into the
rightmost bits. This process is easy to mechanize to produce
the antilog table by computer rather than by hand (which can be tedious
and mistake-prone).
» pip install GF256