# Orders of Gauss Periods in Finite Fields

@article{Gathen1998OrdersOG,
title={Orders of Gauss Periods in Finite Fields},
author={Joachim von zur Gathen and Igor E. Shparlinski},
journal={Applicable Algebra in Engineering, Communication and Computing},
year={1998},
volume={9},
pages={15-24}
}
• Published 4 December 1995
• Mathematics
• Applicable Algebra in Engineering, Communication and Computing
Abstract It is shown that Gauss periods of a special type give an explicit polynomial-time computation of elements of exponentially large multiplicative order in some finite fields. This can be considered as a step towards solving the celebrated problem of finding primitive roots in finite fields in polynomial time.
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
Multiplicative order of Gauss periods
• Mathematics, Computer Science
• 2007
We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.
Gauss periods: orders and cryptographical applications
• Computer Science, Mathematics
Math. Comput.
• 1998
Results indicate that Gauss periods have high order and are often primitive (self-dual) normal elements in finite fields and it is shown thatGauss periods can be exponentiated in quadratic time.
Elements of large order on varieties over prime finite fields
• Mathematics
• 2014
Let V be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C(V) such that for almost all primes p for all but at most
Normal high order elements in finite field extensions based on the cyclotomic polynomials
• Mathematics
Algebra and Discrete Mathematics
• 2020
We consider elements which are both of high multiplicative order and normal in extensions $$F_{q^{m} }$$ of the field $$F_{q}$$. If the extension is defined by a cyclotomic polynomial, we construct
Sharpening of the Explicit Lower Bounds for the Order of Elements in Finite Field Extensions Based on Cyclotomic Polynomials
We explicitly construct elements of high multiplicative order in any extensions of finite fields based on cyclotomic polynomials.
On elements of high order in general finite fields
We show that the Gao’s construction gives for any finite field Fqn elements with the multiplicative order at least ( n+t−1 t ) ∏t−1 i=0 1 d , where d = ⌈
Elements of high order in Artin-Schreier extensions of finite fields $\mathbb F_q$
• Mathematics
• 2015
In this article, we find a lower bound for the order of the coset x+b in the Artin-Schreier extension $\mathbb F_q[x]/(x^p-x-a)$, where $b\in\mathbb F_q$ that satisfies a generic special condition.
Constructing high order elements through subspace polynomials
• Mathematics, Computer Science
SODA
• 2012
This paper presents an algorithm that for any positive integer c and prime power q, finding an element of order exp(Ω(√qc)) in the finite field [EQUATION] in deterministic time (qc)O(1) is easy.