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I have trouble finding a matrix for mapping the members of isomorphism finite fields. For example, for two $GF(2^8)$s constructed from two distinct irreducible polynomials of degree 8, is there any matrix mapping the corresponding members of two fields to each other? Is there any software able to compute such matrix?

Any answer would be appreciated.

Parcly Taxel
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Khalesi
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  • To construct an isomorphism the simplest thing is to find roots of one defining polynomial in the field constructed using the other. A computer program can easily do that by the brute force method of trying out all the alternatives in a small field like this. I wouldn't want to do it by hand though (unless one of my quick tricks of substituting $x\to x+1$ or going reciprocal works - unlikely as those tricks only cover six out of thirty irreducible octics). – Jyrki Lahtonen Aug 20 '17 at 07:34
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    Anyway, yes, after you find a single such zero, you can easily produce the required matrix. If $\alpha$ and $\beta$ are zeros of the two defining polynomials, and $p(\beta)$ is a zero of the minimal polynomial of $\alpha$ in $GF(2)(\beta)$, then the isomorphism maps $\alpha^i\mapsto p(\beta)^i$, $i=0,1,2,\ldots,7$. After that you can just build the matrix using coefficients of $p(\beta)^i$ as columns. – Jyrki Lahtonen Aug 20 '17 at 07:37
  • I suggest to see Example2.3 on page 27 of the outstanding book guide to elliptic curve cryptography by hankerson menezes and vanstone – Amin235 Aug 21 '17 at 08:04

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