Is there any fast way to check if the following equation holds?
$x^{2^q}-x$ mod $p(x)=0$
Polynomials are over finite field $GF(2^q)$
I am aware of the algorithm which uses repeated squaring. This algorithm can achieve a complexity of $O(log(2^q))$.
The above mentioned algorithm actually first calculates $x^{2^q}$ mod $p(x)$, and then compare it with $x$. However, since I only care about if $x^{2^q}=x$ mod $p(x)$. That is, I do not have to know what $x^{2^q}$ mod $p(x)$ is. I was thinking if there exists an algorithm that can solve this problem faster.
Thanks