Is there any algorithm that produces irreducible polynomials over $\mathbb{F}_2$?

Question

Is there any (fast) algorithm that produces irreducible polynomials over $\mathbb{F}_2$?

EDIT: I look up for a irreducible polynomial generator, that is different from decider algorithm for irreducibility.

EDIT2: By Input $n \in \mathbb{N}$ to the algorithm $A$ we expect $n$ irreducible polynomials over $\mathbb{F}_2$. I need that $A$ always returns same result by a specific input $n$ (I don't want random irr. polynomials generator).

Answer 1

A) If you come across an algorithm that produces minimal polynomials, it will fit. I am not aware of such an algo.

B) Nor sure how fast the following is...

1) $x^2 + x + 1$ is the only irreducible poly of $deg=2$
2) $x^3 + x + 1$ and $x^3 + x^2 + 1$ - all irreducible of $deg=3$
3) for any (next) degree $k$, pick a poly $f$ of degree $k$, try to divide it by $x$, $x+1$ and all irreducible of degree up to $[k/2]$. If no divisor is found, $f$ is irreducible.
4) For your algo to produce the same result, return the first $n$ irreducible polys (of low degrees).