I need to know what are the efficient algorithms to find all the irreducible polynomials of a given degree, say $d$ over a given finite field, say $\mathbb{F}\_{p^n}.$


One way is to factorize the polynomial $x^{p^{dn}}-x$, which is the product of all irreducible polynomials whose degree divides $d$, using factorization algorithms and collect all the degree $d$ factors. But I guess we are doing some extra job here. Are there better algorithms to find all irreducible polynomials of degree $d$ ?


I also want to know about the algorithms to find one irreducible polynomial of a given degree over a given finite field.



Algorithms to find irreducible polynomials of a given degree
The last word on the second question is this [paper of Couveignes and Lercier.](http://arxiv.org/abs/0905.1642) The question is highly nontrivial.


