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Actually, you can do Shamir Secret Sharing over any finite field $GF(p^k)$, for any prime $p$ and any integer $k$. If $k=1$, you have the $GF(p)$ field you mentioned; however it works on extension fields as well. We often pick $p=2$ and $k$ a multiple of 8; this makes everything nice even number of bytes (at the cost of doing our calculations in $GF(2^k)$). ...


As I have duplicated your question by mistake and non of us have had an answer, I request help to the authors to know why those parameters where selected like that. The affine transformation is a vector space operation $(\mathbb{F}_{2})^8$, and the simplicity comes from the fact that, from the bunch of possible transformations the one used can be also ...

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