On Finding Primitive Roots in Finite Fields

On Solving Univariate Polynomial Equations over Finite Fields and Some Related Problems

We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorith...

Ducci-sequences and cyclotomic polynomials

We study Ducci-sequences using basic properties of cyclotomic polynomials over F2. We determine the period of a given Ducci-sequence in terms of the order of a polynomial, and in terms of the multiplicative orders of certain elements in finite fields. We also compute some examples and study links between Duccisequences, primitive polynomials and Artin’s conjecture on primitive roots. © 2005 Els...

Cyclotomic Swan subgroups and primitive roots

On Primitive Roots of Tori: The Case of Function Fields

We generalize Bilharz’s Theorem for Gm to all one-dimensional tori over global function fields of finite constant field. As an application, we also derive an analogue, in the setting of function fields, of a theorem (Chen-Kitaoka-Yu, Roskam) on the distribution of fundamental units modulo primes.

On a Question of Davenport and Lewis on Character Sums and Primitive Roots in Finite Fields

Let χ be a nontrivial multiplicative character of Fpn . We obtain the following results related to Davenport-Lewis’ paper [DL] and the Paley Graph conjecture. (1). Let ε > 0 be given. If