Existence of primitive 2-normal elements in finite fields

Aguirre, Josimar J. R.; Neumann, Víctor

doi:10.1016/j.ffa.2021.101864

Cited by 13 publications

(5 citation statements)

References 13 publications

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“…Proposition 3.7. Let N be a positive integer such that 1 2 − 1 N − log q 2 > 0 and let e be a positive integer such that 1 N m ≥ e log 2 log Pe and q n −1 r i ≥ P e for all i ∈ {1, . .…”

Section: General Resultsmentioning

confidence: 99%

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About r-primitive and k-normal elements in finite fields

Aguirre

Carvalho

Neumann

2022

Des. Codes Cryptogr.

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Let F q n be a finite field with q n elements and r be a positive divisor of q n − 1. An element α ∈ F * q n is called r-primitive if its multiplicative order is (q n −1)/r. Also, α ∈ F q n is k-normal over F q if the greatest common divisor of the polynomials g α (x) = αx n−1 +α q x n−2 +. . .+α q n−2 x+α q n−1 and x n −1 in F q n [x] has degree k. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers m 1 , m 2 , k 1 , k 2 , positive integers r 1 , r 2 and rational functions F (x) = F 1 (x)/F 2 (x) ∈ F q n (x) with deg(F i ) ≤ m i for i ∈ {1, 2} satisfying certain conditions and we present sufficient conditions for the existence of r 1 -primitive k 1 -normal elements α ∈ F q n over F q , such that F (α) is an r 2 -primitive k 2 -normal element over F q . Finally as an example we study the case where r 1 = 2, r 2 = 3, k 1 = 2, k 2 = 1, m 1 = 2 and m 2 = 1, with n ≥ 7.

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Section: General Resultsmentioning

confidence: 99%

“…In [23], the authors worked out the case k = 1, and established a Primitive 1-Normal Element Theorem. In [1], the authors showed conditions for the existence of primitive 2normal elements. Generalizing these ideas, in [2] and [21] some conditions for the existence of r-primitive, k-normal elements in F q n over F q were obtained.…”

Section: Introductionmentioning

confidence: 99%

About r-primitive and k-normal elements in finite fields

Aguirre

Carvalho

Neumann

2022

Des. Codes Cryptogr.

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“…In [8,Lemma 3.1] the author gives a way to construct k-normal elements from a given normal element and in [1,Lemma 3.1] the authors show a numerical condition of the existence of 2-normal elements.…”

Section: Introductionmentioning

confidence: 99%

Number of $k$-normal elements over a finite field

Aguirre¹,

Neumann²

2022

Preprint

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An element α ∈ F q n is a normal element over F q if the conjugates α q i , 0 ≤ i ≤ n − 1, are linearly independent over F q . Hence a normal basis for F q n over F q is of the form {α, α q , . . . , α q n−1 }, where α ∈ F q n is normal over F q .In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of knormal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of k-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).

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“…Other problems in this direction concern the existence of a set with some prescribed property whose elements are all primitives (see e.g. [1,3,4,5,6,7,8,12,14]).…”

Section: Introductionmentioning

confidence: 99%

On arithmetic progressions in finite fields

Lemos¹,

Neumann²,

Ribas³

2022

Preprint

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In this paper, we explore the existence of m-terms arithmetic progressions in F q n with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for m ≥ 4 and concrete results for m ∈ {2, 3}, where the complete list of exceptions when the common difference belongs to F * q is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.

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Existence of primitive 2-normal elements in finite fields

Cited by 13 publications

References 13 publications

About r-primitive and k-normal elements in finite fields

About r-primitive and k-normal elements in finite fields

Number of $k$-normal elements over a finite field

On arithmetic progressions in finite fields

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