Fast construction of irreducible polynomials over finite fields

We present a randomized algorithm that on inputting a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d (1+E >(d))x(log q)(5+E >(q)) elementary operations. The function E > in this expression is a real positive function belonging to the class o(1), especially, the complexity is quasi-linear in the degree d. Once given such an irreducible polynomial of degree d, we can compute random irreducible polynomials of degree d at the expense of d (1+E >(d)) x (log q)(1+E >(q)) elementary operations only.