I am interested in an isomorphism between two multiplicative groups of quotient rings $GF(2)[x]/p(x)$, specifically an algorithmic method of determining a mapping between elements.
One is the group $G_1$ of all elements $x^{jk} \bmod p_1(x)$ where $p_1(x)$ is a primitive polynomial of degree $n$, and $j$ is an integer such that $1 < j < m = 2^n-1$ and $j | m$.
The other is a group $G_2$ of all elements $x^k \bmod p_2(x)$ where $p_2(x)$ is a non-primitive polynomial of degree $n$, and the order of the group $G_2$ is $(2^n-1)/j$.
For example, consider $p_1(x) = x^8 + x^4 + x^3 + x^2 + 1$ and $j=15$, where group $G_1$ has elements $1$, $x^{15} = x^5 + x^2 + x$, $x^{30} = x^6 + x^5$, etc., and $p_2(x) = x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$, where group $G_2$ has elements $1$, $x$, $x^2$, $x^3$, $x^4$, $x^5$, $x^6$, $x^7$, $x^7 + x^6 + x^4 + x^2 + x + 1$, etc.
Both groups are cyclic with order 17, but I don't know how I could algorithmically calculate a mapping between elements, aside from just taking $x^{jk}$ in group $G_1$ and $x^k$ in group $G_2$ and just creating a table of the two for each value of $k$.
Any suggestions on determining a mapping function for each direction?