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 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.

Counting points of given height that generate a quadratic extension of a function field

2015 ◽  
Vol 11 (02) ◽  
pp. 569-592 ◽  
Author(s):  
David Kettlestrings ◽  
Jeffrey Lin Thunder
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Function Field ◽  
Asymptotic Estimate ◽  
Algebraic Closure ◽  
Quadratic Extension ◽  
Rational Functions ◽  
Rational Numbers ◽  
Algebraic Extension ◽  
Counting Points

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

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

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 The length of the continued fraction expansion for a class of rational functions in

1991 ◽  
Vol 34 (1) ◽  
pp. 7-17 ◽  
Author(s):  
Arnold Knopfmacher
Keyword(s):  
Finite Field ◽  
Continued Fraction ◽  
Rational Functions ◽  
Euclidean Algorithm ◽  
Continued Fraction Expansion ◽  
Partial Quotients ◽  
Fraction Algorithm

A study is made of the length L(h, k) of the continued fraction algorithm for h/k where h and k are co-prime polynomials in a finite field. In addition we investigate the sum of the degrees of the partial quotients in this expansion for h/k, h, k in . The above continued fraction is determined by means of the Euclidean algorithm for the polynomials h, k in .

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 Values of rational functions in small subgroups of finite fields and the identity testing problem from powers

2019 ◽  
Vol 16 (02) ◽  
pp. 219-231
Author(s):  
László Mérai
Keyword(s):  
Finite Field ◽  
Rational Functions ◽  
Algorithmic Problem ◽  
Identity Testing ◽  
Affine Subspaces ◽  
Algorithmic Problems ◽  
Low Dimensional ◽  
High Degree ◽  
Multiplicative Groups ◽  
Oracle Access

Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field [Formula: see text] considered as a linear space over a subfield [Formula: see text]. We apply this to the recently introduced algorithmic problem of identity testing of “hidden” polynomials [Formula: see text] and [Formula: see text] over a high degree extension of a finite field, given oracle access to [Formula: see text] and [Formula: see text].

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