What criteria do we have to efficiently test if a binary polynomial $P(x)$ is irreducible?
Assume $P$ is given as a vector of $n+1$ bits $c_i$, with $c_0=c_n=1$, as $P(x)=\displaystyle\sum_{j=0}^nc_j\,x^j$ .
If $P(x)$ is irreducible, then $x^{2^n-1}\bmod P=1$ (proof: the order of $x$ in the multiplicative group modulo $x$ divides the group order). This can give a quick and certain indication that a polynomial is not irreducible (the test runs in $O(n^3)$ binary operations using $O(n)$ bits of memory with classical algorithms).
Unfortunately, much like the Fermat pseudo-prime test to base 2 has pseudoprimes (OEIS A001567), there are pseudo-primitives to this test. The list to degree 9 inclusive is (not yet at OEIS but soon to become A316970): $$\begin{array}{l} 1+x+x^4+x^6\\ 1+x^2+x^5+x^6\\ 1+x+x^2+x^3+x^4+x^5+x^6\\ 1+x+x^2+x^4+x^8\\ 1+x+x^3+x^4+x^5+x^7+x^8\\ 1+x^4+x^6+x^7+x^8\\ \dots\end{array}$$
How can we make a more selective test? Is there an analog to the strong pseudoprime test?
Update: found the related Rabin's test for polynomial irreducibility over $\Bbb F_2$
