Generalized Combining of Elements From Finite Fields

Elements of High Order on Finite Fields from Elliptic Curves

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that the values on points of order small with respect to their degree of rational functions on an elliptic curve have high order. We discuss several special cases, including an old construction of Wiedemann, giving the first non-trivial estimate for the ...

On Completely Free Elements in Finite Fields

Elements of Provable High Orders in Finite Fields

A method is given for constructing elements in Fqn whose orders are larger than any polynomial in n when n becomes large. As a by-product a theorem on multiplicative independence of compositions of polynomials is proved.

On primitive elements in finite fields of low characteristic

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, Fqn , small q and large n, we show that the set of all low degree polynomials contains the expected number of primitive elements. The main theorem we prove is a bound for character sums over short intervals in function fields. Our result is unconditional and ...

Finding Primitive Elements in Finite Fields of Small Characteristic

We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field with p elements. In time polynomial in p and n, the algorithm either outputs an element that is provably a generator or declares that it has failed in finding one. Under a heuristic assumption, we argue that the algorithm does always succeed in finding a generator. The algorith...