Finding isomorphism between galois field $ {\rm GF}(2^8)$ and polynomial ring of galois field ${\rm GF}(2^4)[X] / F[X]$

Question

How to find an isomorphism between finite Galois field and polynomial ring of Galois field?

For example,

Let $F_1 = GF(2^8)$ where polynomial of $GF(2^8)$ is $a^8 + a^4 + a^3 + a + 1$,

$F_2 = GF(2^4)[X] / (X^2 + b^2X + b^2)$ with polynomial of $GF(2^4)$ is $b^4 + b + 1$.

$(a^8 + a^4 + a^3 + a + 1)$, $(b^4 + b + 1)$, $(X^2 + b^2X + b^2)$ are all irreducible polynomial. So cardinality of $F_1, F_2$ are the same which implies the existance of an isomorphism $\phi : F_1 \rightarrow F_2$. But I don't know how to find a specific isomorphism. I even don't know what keywords to search, so I would appreciate your help.

(Moreover, above example is for the simplicity. The problem I'm currently facing is larger field such as an isomophism between $GF(2^{512}), GF(2^{128})/ f(X)$ where $f(X)$ is irreducible 4th degree polynomial. So I need a generic solution, not only for the above example.)

Answer 1

I can show you how to find the isomorphism in Magma:

> F2 := GF(2);
> PF2<x>:=PolynomialRing(F2);
> F16<b> := ext< F | x^4 + x + 1 >;
> PF16<y> := PolynomialRing(F16);
> F256 := ext< F16 | y^2 + b^2*y + b^2 >;
> F2 := GF(2);                           
> PF2<x>:=PolynomialRing(F2);
> F16<b> := ext< F | x^4 + x + 1 >;
> PF16<y> := PolynomialRing(F16);
> F256<c> := ext< F16 | y^2 + b^2*y + b^2 >;
> PF256<a> := PolynomialRing(F256);
> Factorization(a^8 + a^4 + a^3 + a + 1);
[
    <a + c^95, 1>,
    <a + c^125, 1>,
    <a + c^175, 1>,
    <a + c^190, 1>,
    <a + c^215, 1>,
    <a + c^235, 1>,
    <a + c^245, 1>,
    <a + c^250, 1>
]

So you can define an isomorphism by mapping $a$% onto any of $-c^{95}$, $-c^{125}$, etc.

Going in the other direction, we can calculate the minimal polynomial of $c$ of F2:

> MinimalPolynomial(c,GF(2));
x^8 + x^5 + x^3 + x^2 + 1

which is similar to that of $a$ but not the same.