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I am confused on this problem. I am not sure what is happening with the (-3) for it to become (12), to then become a larger polynomial in a GF type of problem.

Here is more information about the problem. I hope it helps, but I don't have an actual problem per se. It's more of a concept I want to understand.

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Can someone explain why they are doing that calculation with the multiplicative inverses please?

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Arthur
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J. Doe Hue
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1 Answers1

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We are in $GF(2^n)$, with $n=4$. It is the splitting field of $x^{16}-x$. Thus $g^{15}=1$ (or $g^{-3}=g^{12}$).

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    One doesn't actually need the notion of splitting field for this. In a field of $q$ elements, the multiplicative group of non-zero elements has $q-1$ elements. So by Lagrange's theorem from group theory, all its elements satisfy $x^{q-1}=1$. – Andreas Blass Dec 10 '18 at 21:49
  • @AndreasBlass Nice. Thanks for that insight. –  Dec 10 '18 at 21:59
  • Thanks for the explanation. – J. Doe Hue Dec 10 '18 at 22:05