scholarly journals GENERATORS OF FINITE FIELDS WITH PRESCRIBED TRACES

2021 ◽  
pp. 1-12
Author(s):  
LUCAS REIS ◽  
SÁVIO RIBAS
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Special Class ◽  
Existence Results ◽  
Primitive Elements ◽  
Field Extensions ◽  
Intermediate Field

Abstract This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.

scholarly journals On the multiplicative order of elements in Wiedemann's towers of finite fields

2015 ◽  
Vol 7 (2) ◽  
pp. 220-225
Author(s):  
R. Popovych
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Field Extensions ◽  
Fermat Numbers ◽  
Multiplicative Order ◽  
Order Of Elements

We consider recursive binary finite field extensions $E_{i+1} =E_{i} (x_{i+1} )$, $i\ge -1$, defined by D. Wiedemann. The main object of the paper is to give some proper divisors of the Fermat numbers $N_{i} $ that are not equal to the multiplicative order $O(x_{i} )$.

scholarly journals Existence results for primitive elements in cubic and quartic extensions of a finite field

2018 ◽  
Vol 88 (316) ◽  
pp. 931-947 ◽  
Author(s):  
Geoff Bailey ◽  
Stephen D. Cohen ◽  
Nicole Sutherland ◽  
Tim Trudgian
Keyword(s):  
Finite Field ◽  
Existence Results ◽  
Primitive Elements

scholarly journals FINITE FIELD EXTENSIONS WITH THE LINE OR TRANSLATE PROPERTY FOR -PRIMITIVE ELEMENTS

2020 ◽  
pp. 1-7 ◽  
Author(s):  
STEPHEN D. COHEN ◽  
GIORGOS KAPETANAKIS
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Primitive Elements ◽  
Field Extensions ◽  
Multiplicative Order

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^{n}-1$ . We say that the extension $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements property if, for every $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that $\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$ , there exists some $x\in \mathbb{F}_{q}$ such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$ . We prove that, for sufficiently large prime powers $q$ , $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements. We also discuss the (weaker) translate property for extensions.

scholarly journals Primitive element pairs with a prescribed trace in the quartic extension of a finite field

2020 ◽  
pp. 2150168
Author(s):  
Stephen D. Cohen ◽  
Anju Gupta
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Primitive Element ◽  
Field Extensions

In this paper, we give a largely self-contained proof that the quartic extension [Formula: see text] of the finite field [Formula: see text] contains a primitive element [Formula: see text] such that the element [Formula: see text] is also a primitive element of [Formula: see text] and [Formula: see text] for any prescribed [Formula: see text]. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl. 54 (2018) 1–14.].

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

scholarly journals Linear combinations of primitive elements of a finite field

2018 ◽  
Vol 51 ◽  
pp. 388-406 ◽  
Author(s):  
Stephen D. Cohen ◽  
Tomás Oliveira e Silva ◽  
Nicole Sutherland ◽  
Tim Trudgian
Keyword(s):  
Finite Field ◽  
Primitive Elements ◽  
Linear Combinations

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

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