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 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).

Answer from Dilip Sarwate on Stack Exchange
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polynomials - How to evaluate GF(256) element - Mathematics Stack Exchange › questions › 1843177 › how-to-evaluate-gf256-element
I am assuming that the primitive polynomial is $D^8+D^4+D^3+D^2+1$. For example for GF$(8)$ what we do is as follow to calculate $\alpha^3$ is divide it by $\alpha^3+\alpha+1$ and we get $\alpha+1$ but here in GF$(256)$ this will be really tedious so I would like to know is there any way to ...
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Package gf256 implements arithmetic over Galois Fields with characteristic 2 and exponent 8.