I am trying to construct a Galois field $GF(2^{10})$ from the field $GF(2^5)$ by making a composite field $GF((2^5)^2)$. The method I am using is to find a primitive polynomial of the field $GF(2^{10})$ $x^2+x+p_0$ with ground field $GF(2^5)$. However, I am not having success when the primitive polynomial $GF(2^{10})$ is $x^{10}+x^3+1$. None of the 32 possible values of $p_0$ work. Am I doing something wrong or is it just impossible to achieve what I want for a specific primitive polynomial?
What I am eventually trying to accomplish is to implement the composite Galois field multiplier described on pp. 8-10 of this paper for $GF(2^{10})$ but I cannot find the value of $p_0$.