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 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 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 Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

Construction of MDS Matrices Based on the Primitive Elements of the Finite Field

2021 ◽  
Author(s):  
Wu You ◽  
Dong Xin-feng ◽  
Wang Jin-bo ◽  
Zhang Wen-zheng
Keyword(s):  
Finite Field ◽  
Primitive Elements

On special pairs of primitive elements over a finite field

2021 ◽  
Vol 73 ◽  
pp. 101839
Author(s):  
Cícero Carvalho ◽  
João Paulo Guardieiro ◽  
Victor G.L. Neumann ◽  
Guilherme Tizziotti
Keyword(s):  
Finite Field ◽  
Primitive Elements

scholarly journals Primitive values of rational functions at primitive elements of a finite field

2021 ◽  
Vol 219 ◽  
pp. 237-246
Author(s):  
Stephen D. Cohen ◽  
Hariom Sharma ◽  
Rajendra Sharma
Keyword(s):  
Finite Field ◽  
Rational Functions ◽  
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.

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