Probabilistic Construction of Normal Basis. (Note)

Let Fq be the finite field with q elements. A normal basis polynomial f 2 Fq(x) of degree n is an irreducible polynomial, whose roots form a (normal) basis for the field extension Fqn : Fq. We show that a normal basis polynomial of degree n can be found in expected time O(n2+ǫ · log(q) + n3+ǫ), when an arithmetic operation and the generation of a random constant in the field Fq cost unit time. Given some basis B = {®1,®2,...,®n} for the field extension Fqn : Fq together with an algorithm for multiplying two elements in the B- representation in time O(nβ), we can find a normal basis for this extension and express it in terms of B in expected time O(n1+β+ǫ · log(q) + n3+ǫ).