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How do we find multiplicative inverses using Fermat's Little Theorem? I have the basic understanding of how we compute inverses of numbers, but can someone help with calculating inverses of polynomials in $$GF(2^8)$$

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marked as duplicate by kelalaka, Maeher, Squeamish Ossifrage, Ella Rose Mar 15 at 16:33

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  • $\begingroup$ I know it won't be as effective as the EEA, but I just want to show in my report that there are two methods we can calculate multiplicative inverses for S-box in AES. And as for your question, the only thing I really know is the theorem itself and how it can be applied to numbers to find their inverses. However, I have no idea how I would go about calculating inverses of polynomials using this method $\endgroup$ – user66581 Mar 11 at 22:18
  • $\begingroup$ I actually read that, it doesn't specify how I would go about calculating inverses for polynomials. $\endgroup$ – user66581 Mar 11 at 22:24
  • $\begingroup$ Fermat's little theorem is that $a^p = a$ in $\operatorname{GF}(p)$ for prime $p$, or equivalently $F(a) = a$ where $F(a) = a^p$ is the Frobenius map. Obviously $F$ is not the identity on $\operatorname{GF}(p^k)$ for $k > 1$, but is there another relation involving $F$ that you could think of as an extension of Fermat's little theorem to finite field extensions? $\endgroup$ – Squeamish Ossifrage Mar 11 at 22:33
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    $\begingroup$ A concise explanation is the second bullet there. $\endgroup$ – fgrieu Mar 12 at 6:04

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