Inverses of $r$ -primitive $k$ -normal elements over finite fields

Mamta Rani mamta11singla@gmail.com Avnish K. Sharma avkush94@gmail.com Sharwan K. Tiwari shrawant@gmail.com Anupama Panigrahi anupama.panigrahi@gmail.com Department of Mathematics, University of Delhi, New Delhi, 110007, India Scientific Analysis Group, Defence Research

\&

Development Organization, Metcalfe House, Delhi, 110054, India

Abstract

Let $r$ , $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^{n}-1.$ An element $\alpha$ of the finite field $\mathbb{F}_{q^{n}}$ is called an $r$ -primitive element, if its multiplicative order is $(q^{n}-1)/r$ , and it is called a $k$ -normal element over $\mathbb{F}_{q}$ , if the greatest common divisor of the polynomials $m_{\alpha}(x)=\sum_{i=1}^{n}\alpha^{q^{i-1}}x^{n-i}$ and $x^{n}-1$ is of degree $k.$ In this article, we define the characteristic function for the set of $k$ -normal elements, and with the help of this, we establish a sufficient condition for the existence of an element $\alpha$ in $\mathbb{F}_{q^{n}}$ , such that $\alpha$ and $\alpha^{-1}$ both are simultaneously $r$ -primitive and $k$ -normal over $\mathbb{F}_{q}$ . Moreover, for $n>6k$ , we show that there always exists an $r$ -primitive and $k$ -normal element $\alpha$ such that $\alpha^{-1}$ is also $r$ -primitive and $k$ -normal in all but finitely many fields $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , where $q$ and $n$ are such that $r\mid q^{n}-1$ and there exists a $k$ -degree polynomial $g(x)\mid x^{n}-1$ over $\mathbb{F}_{q}$ . In particular, we discuss the existence of an element $\alpha$ in $\mathbb{F}_{q^{n}}$ such that $\alpha$ and $\alpha^{-1}$ both are simultaneously $1$ -primitive and $1$ -normal over $\mathbb{F}_{q}$ .

keywords:

Finite fields,

r

-Primitive elements,

k

-Normal elements, Additive and multiplicative characters.

MSC:

[

2020

] 12E20, 11T23.

1 Introduction

Let $\mathbb{F}_{q^{n}}$ be the field extension of degree $n$ over $\mathbb{F}_{q},$ where $q$ be a prime power and $n\in\mathbb{N}$ . We recall that, the multiplicative group $\mathbb{F}_{q^{n}}^{*}$ is cyclic, and an element $\alpha\in\mathbb{F}_{q^{n}}^{*}$ is called primitive, if its multiplicative order is $q^{n}-1.$ Let $r$ be a divisor of $q^{n}-1$ , then an element $\alpha\in\mathbb{F}_{q^{n}}^{*}$ is called $r$ -primitive, if its multiplicative order is $(q^{n}-1)/r$ . Clearly, a 1-primitive element in $\mathbb{F}_{q^{n}}$ is actually a primitive element. Moreover, the existence of $r$ -primitive elements is obvious, because the multiplicative order of the $r^{\mathrm{th}}$ power of a primitive element is $(q^{n}-1)/r$ . Primitive elements have many applications in the field of cryptography [5, 12, 2] due to their highest multiplicative order. Moreover, $r$ -primitive elements can also be considered as high ordered elements for small values of $r$ . Therefore, in many applications, $r$ -primitive elements may replace primitive elements.

Again, recall that, the finite field $\mathbb{F}_{q^{n}}$ is also an $\mathbb{F}_{q}$ -vector space, and an element $\alpha\in\mathbb{F}_{q^{n}}$ is said to be normal over $\mathbb{F}_{q}$ , if the set $\{\alpha,\alpha^{q},\ldots,\alpha^{q^{n-1}}\}$ forms a basis (called normal basis) of $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ . Normal bases have many applications in the computational theory due to their efficient implementation in finite field arithmetic. For more detail on normal elements one may refer [7] and the references therein.

From [11, Theorem 2.39], we know that an element $\alpha\in\mathbb{F}_{q^{n}}$ is normal over $\mathbb{F}_{q}$ if and only if the polynomials $m_{\alpha}(x)=\alpha x^{n-1}+\alpha^{q}x^{n-2}+\ldots+\alpha^{q^{n-2}}x+\alpha^{q^{n-1}}$ and $x^{n}-1$ are relatively prime in $\mathbb{F}_{q^{n}}[x].$ Motivated by this definition, Huczynska et al. [8] introduced the concept of $k$ -normal elements and gave the following definition.

Definition 1.1.

An element $\alpha\in\mathbb{F}_{q^{n}}$ is said to be $k$ -normal over $\mathbb{F}_{q},$ if the polynomials $m_{\alpha}(x)=\alpha x^{n-1}+\alpha^{q}x^{n-2}+\ldots+\alpha^{q^{n-2}}x+\alpha^{q^{n-1}}$ and $x^{n}-1$ has greatest common divisor of degree $k$ over $\mathbb{F}_{q}$ .

From the above definition, it is clear that $0\leq k\leq n-1.$ We call an element $\alpha\in\mathbb{F}_{q^{n}}$ , $r$ -primitive $k$ -normal, if it is both $r$ -primitive and $k$ -normal over $\mathbb{F}_{q}.$ From the above definition, if $k=0$ , then $\alpha$ is normal over $\mathbb{F}_{q}$ . Moreover, Huczynska et al. [8] characterized the $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ and established bounds for the number of such elements, and proved the existence of an element $\alpha$ which is both $1$ -primitive and $1$ -normal over $\mathbb{F}_{q}$ for $n\geq 6$ if $q\geq 11$ , and for $n\geq 3$ if $3\leq q\leq 9,$ where $\mathrm{gcd}(q,n)=1.$ In 1987, Lenstra and Schoof [10] proved the existence of primitive normal elements (i.e. $1$ -primitive $0$ -normal elements) for arbitrary finite fields. In 2014, Anderson and Mullen [13] conjectured that, for a prime $p\geq 5$ and an integer $n\geq 3$ , there always exists an element $\alpha\in\mathbb{F}_{p^{n}}$ of order $(p^{n}-1)/r$ , which is $k$ -normal for $(r,k)=(1,1),(2,0),(2,1)$ . In 2018, Reis and Thomson in [17] settled their conjecture for $(r,k)=(1,1)$ , for arbitrary $q$ and $n\geq 3$ . Finally, in 2019, Kapetanakis and Reis [9] completely settled the above conjecture for arbitrary prime power $q$ .

Recently, Aguirre and Neumann [1] showed that, there exists a $1$ -primitive $2$ -normal element in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ if and only if $n\geq 5$ and $\mathrm{gcd}(q^{3}-q,n)\neq 1$ or $n=4$ and $q\equiv 1\hskip 1.42262pt(\mathrm{mod}\hskip 1.42262pt4).$ In this direction, we [14] generalized the above existence problems to arbitrary $r$ and $k$ , and established a sufficient condition for the existence of an element $\alpha$ in $\mathbb{F}_{q^{n}}$ which is simultaneously $r$ -primitive and $k$ -normal over $\mathbb{F}_{q}.$ Moreover, in particular, we gave the following complete existence of $2$ -primitive $2$ -normal elements.

Theorem 1.1.

Let $q$ be a prime power and $n$ be a positive integer. Then, there always exists a $2$ -primitive $2$ -normal element in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ if and only if $q$ is odd, and either $n\geq 5$ and $\mathrm{gcd}(q^{3}-q,n)\neq 1$ or $n=4$ and $q\equiv 1(\mathrm{mod}\hskip 1.42262pt4)$ .

Notice that, the inverse of an $r$ -primitive element is also $r$ -primitive in $\mathbb{F}_{q^{n}}$ for $r\mid q^{n}-1.$ But this may not be true in case of $k$ -normal elements. For example, if $k=0$ , then there does not exist any normal element $\alpha\in\mathbb{F}_{8}$ over $\mathbb{F}_{2}$ for which $\alpha^{-1}$ is also normal element in $\mathbb{F}_{2^{3}}$ over $\mathbb{F}_{2}$ . In [20], Tian and Qi showed the existence of a primitive normal element $\alpha\in\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ whose inverse is also primitive normal over $\mathbb{F}_{q}$ for all $q$ and $n\geq 32.$ Later, Cohen and Huczynska [3] completely settled this problem by using sieving technique and gave the following result.

Theorem 1.2 (Strong Primitive Normal Basis Theorem).

There exists a $(1-)$ primitive $(0-)$ normal element $\alpha\in\mathbb{F}_{q^{n}};$ $n\geq 2,$ such that $\alpha^{-1}$ is also a primitive normal element over $\mathbb{F}_{q}$ unless $(q,n)$ is one of the pair $(2,3),\ (2,4),\ (3,4),\ (4,3),\ (5,4).$

Motivated by the above theorem, in this article, we discuss the existence of a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ . For this, we first define the characteristic function in Section 3 for the set of $k$ -normal elements, which plays a crucial role in establishing a sufficient condition for the existence of such pairs in Section 4. Moreover, we conclude that, for $n>6k$ , there exists a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in all but finitely many fields $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}.$ In particular, we gave the following result in Section 5.

Theorem 1.3.

Let $q$ be a prime power and $n\geq 5$ be an integer. Then

$(i)$

For $n\geq 7$ and $q\geq 2$ , there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ .
$(ii)$

For $n=5$ , $6$ and $q\geq 2$ such that $\mathrm{gcd}(q,n)=1$ , there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ with the sole genuine exception $(4,5).$

2 Preliminaries

In this section, we recall some definitions and lemmas that are crucial for proving the results of this article.

2.1 Freeness

We begin with the following definition.

Definition 2.1 ( $e$ -free element).

Let $e\mid q^{n}-1$ and $\beta\in\mathbb{F}_{q^{n}}.$ Then $\beta$ is said to be $e$ -free, if $\beta$ is not a $d^{th}$ power in $\mathbb{F}_{q^{n}}$ for any divisor $d>1$ of $e$ . In other words, $\beta$ is $e$ -free if and only if $e$ and $\frac{q^{n}-1}{\mathrm{ord}(\beta)}$ are co-prime, where $\mathrm{ord}(\beta)$ is the multiplicative order of $\beta$ in $\mathbb{F}_{q^{n}}^{*}.$ Clearly, an element $\beta\in\mathbb{F}_{q^{n}}$ is primitive if and only if it is $(q^{n}-1)$ -free.

Let $h(x)=\sum_{i=0}^{m}a_{i}x^{i}\in\mathbb{F}_{q}[x]$ with $a_{m}\neq 0$ . Then, the additive group $\mathbb{F}_{q^{n}}$ forms an $\mathbb{F}_{q}[x]$ - module under the action $h\circ\beta=\sum_{i=0}^{m}a_{i}\beta^{q^{i}}$ . Notice that, $(x^{n}-1)\circ\beta=0$ for all $\beta\in\mathbb{F}_{q^{n}},$ which leads to the following definition.

Definition 2.2.

The $\mathbb{F}_{q}$ -order $($ denoted by $\mathrm{Ord}_{q}(\cdot))$ of an element $\beta\in\mathbb{F}_{q^{n}},$ is the least degree monic divisor $h(x)$ of $x^{n}-1$ such that $h\circ\beta=0$ .

Similar to the $e$ -free elements, we have the following definition for $f$ -free elements.

Definition 2.3.

Let $h(x)\mid x^{n}-1$ and $\beta\in\mathbb{F}_{q^{n}}.$ Then, $\beta$ is called $h$ -free, if $\beta\neq g\circ\gamma$ for any $\gamma\in\mathbb{F}_{q^{n}}$ and for any divisor $g(x)$ of $h(x)$ of positive degree. In other words, $\beta$ is $h$ -free if and only if $h(x)$ and $\frac{x^{n}-1}{\mathrm{Ord}_{q}(\beta)}$ are co-prime. Clearly, an element $\beta\in\mathbb{F}_{q^{n}}$ is normal if and only if it is $(x^{n}-1)$ -free.

In this article, we will use the following definition of $k$ -normal elements.

Definition 2.4.

[8, Theorem 3.2] An element $\beta\in\mathbb{F}_{q^{n}}$ is $k$ -normal over $\mathbb{F}_{q}$ , if and only if the $\mathbb{F}_{q}$ -order of $\beta$ is of degree $n-k$ .

The following lemma provides a way to construct the $k$ -normal elements from a given normal element.

Lemma 2.1.

[16, Lemma 3.1] Let $\beta\in\mathbb{F}_{q^{n}}$ be a normal element over $\mathbb{F}_{q}$ and $g\in\mathbb{F}_{q}[x]$ be a polynomial of degree $k$ such that $g$ divides $x^{n}-1$ . Then $\alpha=g\circ\beta$ is $k$ -normal.

2.2 Characters

Let $\mathfrak{G}$ be a finite abelian group. A character $\chi$ of $\mathfrak{G}$ is a homomorphism from $\mathfrak{G}$ into the multiplicative group of complex numbers of unit modulus. The set of all such characters of $\mathfrak{G}$ , denoted by $\widehat{\mathfrak{G}}$ , forms a multiplicative group and $\mathfrak{G}\cong\widehat{\mathfrak{G}}$ . A character $\chi$ is called the trivial character if $\chi(\alpha)=1$ for all $\alpha\in\mathfrak{G}$ , otherwise it is a non-trivial character.

Lemma 2.2.

[11, Theorem 5.4] Let $\chi$ be any non-trivial character of a finite abelian group $\mathfrak{G}$ and $\alpha\in\mathfrak{G}$ be any non-trivial element, then

\sum_{\alpha\in\mathfrak{G}}\chi(\alpha)=0\ \text{and}\ \sum_{\chi\in\widehat{\mathfrak{G}}}\chi(\alpha)=0.

Let $\psi$ denote the additive character for the additive group $\mathbb{F}_{q^{n}}$ , and $\chi$ denote the multiplicative character for the multiplicative group $\mathbb{F}_{q^{n}}^{*}.$ The additive character $\psi_{0}$ defined by $\psi_{0}(\beta)=e^{2\pi i\mathrm{Tr}(\beta)/p},\ \text{for all}\ \beta\in\mathbb{F}_{q^{n}},$ where $p$ is the characteristic of $\mathbb{F}_{q^{n}}$ and $\mathrm{Tr}$ is the absolute trace function from $\mathbb{F}_{q^{n}}$ to $\mathbb{F}_{p}$ , is called the canonical additive character of $\mathbb{F}_{q^{n}}$ . Moreover, every additive character $\psi_{\beta}$ for $\beta\in\mathbb{F}_{q^{n}}$ can be expressed in terms of the canonical additive character $\psi_{0}$ as $\psi_{\beta}(\gamma)=\psi_{0}(\beta\gamma),\ \text{for all}\ \gamma\in\mathbb{F}_{q^{n}}.$

For any $\psi\in\widehat{\mathbb{F}}_{q^{n}}$ , $\alpha\in\mathbb{F}_{q^{n}}$ and $g(x)\in\mathbb{F}_{q}[x]$ , $\widehat{\mathbb{F}}_{q^{n}}$ is an $\mathbb{F}_{q}[x]$ -module under the action $\psi\circ g(\alpha)=\psi(g\circ\alpha).$ The $\mathbb{F}_{q}$ -order of an additive character $\psi\in\widehat{\mathbb{F}}_{q^{n}}$ , denoted by $\mathrm{Ord}_{q}(\psi)$ , is the least degree monic divisor $g(x)$ of $x^{n}-1$ such that $\psi\circ g$ is the trivial character and there are precisely $\Phi_{q}(g)$ characters of $\mathbb{F}_{q}$ -order $g(x)$ , where $\Phi_{q}(g)=|(\mathbb{F}_{q}[x]/<g>)^{*}|$ . Moreover, $\sum_{h\mid g}\Phi_{q}(h)=q^{\mathrm{deg}(g)}.$

We shall need the following lemma to prove our sufficient condition.

Lemma 2.3.

[6, Theorem 5.5] Let $F(x)=\prod_{j=1}^{k}f_{j}(x)^{n_{j}}$ be a rational function in $\mathbb{F}_{q^{n}}(x)$ , where $f_{j}(x)\in\mathbb{F}_{q^{n}}[x]$ are irreducible polynomials and $n_{j}$ are non-zero integers. Let $\chi$ be a multiplicative character of order $d$ of $\mathbb{F}_{q^{n}}^{*}.$ Suppose that the rational function $F(x)$ is not of the form $L(x)^{d}$ for any $L(x)\in\mathbb{F}(x),$ where $\mathbb{F}$ is the algebraic closure of $\mathbb{F}_{q^{n}}.$ Then we have

\Big{|}\sum_{\begin{subarray}{c}\alpha\in\mathbb{F}_{q^{n}}\\ F(\alpha)\neq 0,\infty\end{subarray}}\chi(F(\alpha))\Big{|}\leq\Big{(}\sum_{j=1}^{k}\mathrm{deg}(f_{j})-1\Big{)}q^{n/2}.

Lemma 2.4.

[6, Theorem 5.6] Let $F(x),\ G(x)\in\mathbb{F}_{q^{n}}(x)$ be rational functions. Write $F(x)=\prod_{j=1}^{k}f_{j}(x)^{n_{j}}$ , where $f_{j}(x)\in\mathbb{F}_{q^{n}}[x]$ are irreducible polynomials and $n_{j}$ are non-zero integers. Let $d_{1}=\sum_{j=1}^{k}\mathrm{deg}(f_{j}(x))$ , $d_{2}=max\{\mathrm{deg}(G(x)),0\}$ , $d_{3}$ be the degree of the denominator of $G(x)$ and $d_{4}$ be the sum of degrees of those irreducible polynomials dividing the denominator of $G(x)$ , but distinct from $f_{j}\ ;\ 1\leq j\leq k.$ Let $\chi$ be the multiplicative character of $\mathbb{F}_{q^{n}}^{*}$ and $\psi$ be the non-trivial additive character of $\mathbb{F}_{q^{n}}.$ Suppose that $G(x)\neq L(x)^{q^{n}}-L(x)$ in $\mathbb{F}(x),$ where $\mathbb{F}$ is the algebraic closure of $\mathbb{F}_{q^{n}}.$ Then we have

\Big{|}\sum_{\begin{subarray}{c}\alpha\in\mathbb{F}_{q^{n}},F(\alpha)\neq 0,\infty\\ G(\alpha)\neq\infty\end{subarray}}\chi(F(\alpha))\psi(G(\alpha))\Big{|}\leq(d_{1}+d_{2}+d_{3}+d_{4}-1)q^{n/2}.

2.3 Some characteristic functions

For the set of $e$ -free elements of $\mathbb{F}_{q^{n}}^{*}$ , we have the following characteristic function $\rho_{e}:\mathbb{F}_{q^{n}}^{*}\to\{0,1\}.$

\rho_{e}(\beta)=\frac{\phi(e)}{e}\sum_{d|e}\frac{\mu(d)}{\phi(d)}\sum_{\chi_{d}}\chi_{d}(\beta),

(1)

where $\mu$ is the Möbius function and the internal sum runs over all the multiplicative characters $\chi_{d}$ of order $d$ . Similar to the characteristic function for the set of $e$ -free elements, we have the following characteristic function $\Upsilon_{g}:\mathbb{F}_{q^{n}}\to\{0,1\}$ for the set of $g$ -free elements of $\mathbb{F}_{q^{n}}$ .

\Upsilon_{g}(\alpha)=\dfrac{\Phi_{q}(g)}{q^{\mathrm{deg}(g)}}\sum_{h|g}\dfrac{\mu^{\prime}(h)}{\Phi_{q}(h)}\sum_{\psi_{h}}\psi_{h}(\alpha),

(2)

where $\mu^{\prime}$ is the analog of the Möbius function, which is defined as $\mu^{\prime}(h)=(-1)^{t}$ , if $h(x)$ is a product of $t$ distinct monic irreducible polynomials, otherwise 0, and the internal sum runs over the additive characters $\psi_{h}$ of the $\mathbb{F}_{q}$ -order $h(x)$ .

3 Characterization of $r$ -primitive and $k$ -normal elements

3.1 Characteristic function for the set of $r$ -primitive elements

Let $r\mid q^{n}-1$ be a positive integer, and write $r=up_{1}^{b_{1}}p_{2}^{b_{2}}\cdots p_{s}^{b_{s}}$ , where $u$ and $(q^{n}-1)/u$ are co-prime and $p_{j}$ ’s are distinct primes such that $b_{j}\geq 1$ and $p_{j}^{b_{j}+1}\mid q^{n}-1$ , for all $j=1,2,\ldots,s.$ In this article, we denote the product of distinct irreducible factors of an integer or a polynomial $m$ by $\mathrm{rad}(m)$ . Now, let $R=\mathrm{rad}(q^{n}-1)/\mathrm{rad}(r)$ , and set $\delta_{j}:=p_{j}^{b_{j}}$ and $\lambda_{j}:=p_{j}^{b_{j}+1}$ for all $j=1,2,\ldots,s$ . In [4], Cohen and Kapetanakis showed that an element $\alpha\in\mathbb{F}_{q^{n}}$ is $r$ -primitive if and only if $\alpha$ is $R$ -free, $\alpha=\beta^{r}$ for some $\beta\in\mathbb{F}_{q^{n}}$ , but $\alpha\neq\beta^{\lambda_{j}}$ for any $\beta\in\mathbb{F}_{q^{n}}$ and any $j=1,2,\ldots,s$ , and using this fact, they defined the characteristic function for $r$ -primitive elements. In [14], following them, we defined the following characteristic function $\Gamma_{r}^{d}$ for the set $Q_{r}^{d}$ of elements $\alpha\in\mathbb{F}_{q^{n}}$ such that $\alpha$ is $d$ -free for some divisor $d$ of $R$ , $\alpha=\beta^{r}$ for some $\beta\in\mathbb{F}_{q^{n}}$ , but $\alpha\neq\beta^{\lambda_{j}}$ for any $\beta\in\mathbb{F}_{q^{n}}$ and any $j=1,2,\ldots,s.$

\Gamma_{r}^{d}(\alpha)=\frac{\phi(d)}{rd}\sum_{\begin{subarray}{c}d_{1}\mid d,d_{2}\mid u\end{subarray}}\sum_{\begin{subarray}{c}e_{j}\mid\lambda_{j},\\ 1\leq j\leq s\end{subarray}}\frac{\mu(d_{1})}{\phi(d_{1})}\big{(}\prod_{j=1}^{s}\ell_{p_{j},e_{j}}\big{)}\sum_{\begin{subarray}{c}\chi_{d_{1}},\chi_{d_{2}},\\ \chi_{e_{1}},\ldots,\chi_{e_{s}}\end{subarray}}(\chi_{d_{1}}\chi_{d_{2}}\chi_{e_{1}}\cdots\chi_{e_{s}})(\alpha),

(3)

where $\ell_{p_{j},e_{j}}=\left\{\begin{array}[]{ll}1-1/p_{j}&\mathrm{if}\ e_{j}\neq\lambda_{j},\\ -1/p_{j}&\mathrm{if}\ e_{j}=\lambda_{j}\end{array}\right..$ In particular, for $d=R$ , the above characteristic function provides the characteristic function given by Cohen and Kapetanakis [4] for the set $Q_{r}^{R}$ of $r$ -primitive elements.

3.2 Characteristic function for the set of $k$ -normal elements

For the existence of those elements in a finite field $\mathbb{F}_{q^{n}}$ that simultaneously satisfy certain properties, such as primitivity, normality, trace, norm, etc., a general approach is to use the characteristic functions for the set of elements with those properties. In this article, we also use this approach to discuss the existence of a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ . For this, we shall need the characteristic function for the set of $k$ -normal elements that we define in this subsection.

Let $S_{k}$ be the collection of all $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ and $P_{k}$ be the collection of polynomials of degree $k$ that divides $x^{n}-1$ . By Lemma 2.1, if $\beta$ is a normal element of $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , then for any polynomial $g\in P_{k}$ , $g\circ\beta$ is $k$ -normal over $\mathbb{F}_{q}$ . So, for every $g\in P_{k}$ , we consider a set $S_{g,k}:=\{\alpha\in S_{k}\ |\ \alpha=g\circ\beta\text{ for some element}\ \beta\in\mathbb{F}_{q^{n}}\}$ of $k$ -normal elements. We have the following lemma.

Lemma 3.1.

If $g,\ g^{\prime}$ are two distinct polynomials in $P_{k}$ , then $S_{g,k}\bigcap S_{g^{\prime},k}=\emptyset$ , and $\bigcup_{g\in P_{k}}S_{g,k}=S_{k}$ .

Proof.

Suppose that, $S_{g,k}\bigcap S_{g^{\prime},k}\neq\emptyset$ , then there exist elements $\beta$ and $\beta^{\prime}$ in $\mathbb{F}_{q^{n}}$ such that $g\circ\beta=g^{\prime}\circ\beta^{\prime}$ , and by uniqueness of the $\mathbb{F}_{q}$ -order of an element, we get $g=g^{\prime}$ , a contradiction. Hence, $S_{g,k}\bigcap S_{g^{\prime},k}=\emptyset$ .

Clearly, $\bigcup_{g\in P_{k}}S_{g,k}\subseteq S_{k}.$ Now, let $\alpha\in S_{k}$ , then $\alpha=h\circ\beta$ , for some normal element $\beta$ and a polynomial $h\in\mathbb{F}_{q}[x]$ of degree at most $n-1$ . Let $f$ be the $\mathbb{F}_{q}$ -order of $\alpha$ , then $f\circ\alpha=fh\circ\beta=0$ , and hence, $x^{n}-1\mid fh$ , i.e. $h=(x^{n}-1)h^{\prime}/f$ , for some $h^{\prime}\in\mathbb{F}_{q}[x].$ Thus, $\alpha=\frac{x^{n}-1}{f}\circ(h^{\prime}\circ\beta)$ , which implies $\alpha=g\circ\gamma\in S_{g,k}$ , where $g=\frac{x^{n}-1}{f}\in P_{k}$ and $\gamma=h^{\prime}\circ\beta\in\mathbb{F}_{q^{n}}$ . Hence, $\bigcup_{g\in P_{k}}S_{g,k}=S_{k}$ . ∎

From the above lemma, it is clear that, the set $\{S_{g,k}\ |\ g\in P_{k}\}$ provides a partition of $S_{k}$ . Therefore, the characteristic function for the set $S_{k}$ of $k$ -normal elements is equal to the sum of the characteristic functions for the sets $S_{g,k}\hskip 1.42262pt;\ g\in P_{k}.$ Hence, it is enough to define the characteristic function for the set $S_{g,k}$ .

Let $g\in P_{k}$ be an arbitrary but fixed polynomial. Write $g=\pi f_{1}^{b_{1}}f_{2}^{b_{2}}\cdots f_{t}^{b_{t}}$ , where $\pi$ and $(x^{n}-1)/\pi$ are co-prime, and $f_{i}$ ’s are distinct irreducible polynomials such that $b_{i}\geq 1$ and $f_{i}^{b_{i}+1}\mid x^{n}-1$ , for all $i=1,2,\ldots,t.$ Further, let $G=\mathrm{rad}(x^{n}-1)/\mathrm{rad}(g)$ , and set $\Delta_{i}:=f_{i}^{b_{i}}$ and $\Lambda_{i}:=f_{i}^{b_{i}+1}$ for all $i=1,2,\ldots,t.$ Now, we prove the following lemma.

Lemma 3.2.

Let $k$ be a non-negative integer and $g\in P_{k}$ . Then, the following are equivalent.

$(i)$

$\alpha\in S_{g,k}$ .
$(ii)$

$\alpha$ is $G$ -free, $\alpha=g\circ\beta$ for some $\beta\in\mathbb{F}_{q^{n}}$ , but $\alpha\neq\Lambda_{i}\circ\beta$ for any $\beta\in\mathbb{F}_{q^{n}}$ and for any $i=1,2,\ldots,t.$

Proof.

First, let $\alpha\in S_{g,k}$ . Then by definition of $S_{g,k}$ , $\alpha=g\circ\beta$ for some $\beta\in\mathbb{F}_{q^{n}}$ , and $\mathbb{F}_{q}$ -order of $\alpha$ is $(x^{n}-1)/g$ . Notice that, $G$ and $g$ are co-prime, therefore, $\mathrm{gcd}\big{(}G,\frac{x^{n}-1}{\mathrm{Ord}_{q}(\alpha)}\big{)}=1$ , which implies $\alpha$ is $G$ -free. Now, suppose that $\alpha=\Lambda_{i}\circ\beta$ for some $\beta\in\mathbb{F}_{q^{n}}$ and some $i=1,2,\ldots,t$ . Then, $\frac{x^{n}-1}{\Lambda_{i}}\circ\alpha=0$ , which implies $\frac{x^{n}-1}{g}\mid\frac{x^{n}-1}{\Lambda_{i}}$ , i.e. $\Lambda_{i}\mid g,$ which is not possible. Hence, $(i)$ implies $(ii)$ .

Now, let $\alpha$ is $G$ -free, $\alpha=g\circ\beta$ for some $\beta\in\mathbb{F}_{q^{n}}$ , but $\alpha\neq\Lambda_{i}\circ\beta$ for any $\beta\in\mathbb{F}_{q^{n}}$ and for any $i=1,2,\ldots,t.$ Then, $\alpha=g\circ(h\circ\gamma)$ for some normal element $\gamma\in\mathbb{F}_{q^{n}}$ and a polynomial $h\in\mathbb{F}_{q}[x]$ of degree at most $n-1$ , and $\mathrm{Ord}_{q}(\alpha)=\frac{x^{n}-1}{g\hskip 1.42262pt\mathrm{gcd}(h,(x^{n}-1)/g)}$ . Suppose that, $h^{\prime}$ be an irreducible factor of the $\mathrm{gcd}(h,(x^{n}-1)/g)$ , then $h^{\prime}\mid h$ and $h^{\prime}\mid\frac{x^{n}-1}{g}$ . This implies, either $h^{\prime}\mid G$ or $\Lambda_{i}\mid gh.$ In any case, we get a contradiction. This means $\mathrm{Ord}_{q}(\alpha)=\frac{x^{n}-1}{g}$ , i.e. $\alpha$ is a $k$ -normal element. Hence, $(ii)$ implies $(i)$ . ∎

Now, we define the characteristic function $\Psi_{g}$ for the set of elements of the form $g\circ\beta.$ Consider the set $M_{g}$ of the elements of the form $g\circ\beta$ , where $g\mid x^{n}-1.$ Clearly, $M_{g}$ is a subgroup of the additive group $\mathbb{F}_{q^{n}}$ . Let $A\subseteq\widehat{\mathbb{F}}_{q^{n}}$ be the annihilator of $M_{g}$ , i.e. the collection of all additive characters $\psi\in\widehat{\mathbb{F}}_{q^{n}}$ such that $\psi(\alpha)=1$ for all $\alpha\in M_{g}.$ Clearly, the set $A$ consists of all the additive characters of $\mathbb{F}_{q}$ -order $h$ dividing $g$ , and from [11, Theorem 5.6], it is isomorphic to the group of the characters of the quotient group $\mathbb{F}_{q^{n}}/M_{g}$ , i.e. $A\cong\widehat{(\mathbb{F}_{q^{n}}/M_{g})}$ . Therefore, we can define a character $\mathcal{Y}_{h}$ on $\widehat{(\mathbb{F}_{q^{n}}/M_{g})}$ by $\mathcal{Y}_{h}(\alpha+M_{g})=\psi_{h}(\alpha),$ where $\psi_{h}\in A.$ Then,

\sum_{h\mid g}\sum_{\mathcal{Y}_{h}}\mathcal{Y}_{h}(\alpha+M_{g})=\left\{\begin{array}[]{lll}q^{\mathrm{deg}(g)}&,&\mathrm{if}\ \alpha\in M_{g},\\ 0&,&\mathrm{otherwise}\end{array}\right..

Thus, we can define a characteristic function $\Psi_{g}:\mathbb{F}_{q^{n}}\to\{0,1\}$ for the set $M_{g}$ as follows.

\Psi_{g}(\alpha):=\frac{1}{q^{\mathrm{deg}(g)}}\sum_{h\mid g}\sum_{\psi_{h}}\psi_{h}(\alpha).

(4)

Now, we are ready to define a characteristic function $\mathcal{Q}_{g}^{G}$ for the set $S_{g,k}.$ From the characteristic functions (2) and (4) and from Lemma 3.2, we have

\displaystyle\mathcal{Q}_{g}^{G}(\alpha)=\Upsilon_{G}(\alpha)\Psi_{g}(\alpha)\prod_{i=1}^{t}\big{(}1-\Psi_{\Lambda_{i}}(\alpha)\big{)}.

Since, $g=\pi\prod_{i=1}^{t}\Delta_{i}$ , and $\pi$ , $\Delta_{i}\ ;\ i=1,2,\ldots,t$ are mutually co-prime, $\Psi_{g}(\alpha)=\Psi_{\pi}(\alpha)\prod_{i=1}^{t}\Psi_{\Delta_{i}}(\alpha)$ , and hence,

\displaystyle\mathcal{Q}_{g}^{G}(\alpha)=\Upsilon_{G}(\alpha)\Psi_{\pi}(\alpha)\prod_{i=1}^{t}\Psi_{\Delta_{i}}(\alpha)\big{(}1-\Psi_{\Lambda_{i}}(\alpha)\big{)}.

Moreover, if $\alpha=\Lambda_{i}\circ\beta$ for some $\beta$ , then $\alpha=\Delta_{i}\circ\gamma$ for some $\gamma$ in $\mathbb{F}_{q^{n}}$ , and $\Psi_{\Delta_{i}}\Psi_{\Lambda_{i}}=\Psi_{\Lambda_{i}}$ for all $i=1,2,\ldots,t.$ Hence,

\mathcal{Q}_{g}^{G}(\alpha)=\Upsilon_{G}(\alpha)\Psi_{\pi}(\alpha)\prod_{i=1}^{t}\big{(}\Psi_{\Delta_{i}}(\alpha)-\Psi_{\Lambda_{i}}(\alpha)\big{)}.

(5)

Now,

	$\displaystyle\Psi_{\Delta_{i}}(\alpha)-\Psi_{\Lambda_{i}}(\alpha)=$	$\displaystyle\frac{1}{q^{\mathrm{deg}(\Delta_{i})}}\sum_{h\mid\Delta_{i}}\sum_{\psi_{h}}\psi_{h}(\alpha)-\frac{1}{q^{\mathrm{deg}(\Lambda_{i})}}\sum_{h\mid\Lambda_{i}}\sum_{\psi_{h}}\psi_{h}(\alpha)$
	$\displaystyle=$	$\displaystyle\frac{1}{q^{\mathrm{deg}(\Delta_{i})}}\sum_{h\mid\Lambda_{i}}\sum_{\psi_{h}}\ell_{f_{i},h}^{\prime}\psi_{h}(\alpha),$

where $\ell_{f_{i},h}^{\prime}=\left\{\begin{array}[]{ll}1-1/q^{\mathrm{deg}(f_{i})}&\mathrm{if}\ h\neq\Lambda_{i},\\ -1/q^{\mathrm{deg}(f_{i})}&\mathrm{if}\ h=\Lambda_{i}\end{array}\right..$ Notice that, $|\ell_{f_{i},h}^{\prime}|\leq\ell_{f_{i},1}^{\prime}$ for all $i=1,2,\ldots,t$ . Now, substituting the values of $\Upsilon_{G}(\alpha)$ , $\Psi_{\pi}(\alpha)$ and $\Psi_{\Delta_{i}}(\alpha)-\Psi_{\Lambda_{i}}(\alpha)$ in Equation (5), we get

\displaystyle\mathcal{Q}_{g}^{G}(\alpha)=

\displaystyle\frac{\Phi_{q}(G)}{q^{\mathrm{deg}(G)+\mathrm{deg}(g)}}\sum_{\begin{subarray}{c}g_{1}\mid G,\\ g_{2}\mid\pi\end{subarray}}\sum_{\begin{subarray}{c}h_{i}\mid\Lambda_{i},\\ 1\leq i\leq t\end{subarray}}\frac{\mu^{\prime}(g_{1})}{\Phi_{q}(g_{1})}\big{(}\prod_{i=1}^{t}\ell_{f_{i},h_{i}}^{\prime}\big{)}\sum_{\begin{subarray}{c}\psi_{g_{1}},\psi_{g_{2}},\\ \psi_{h_{i}};\ 1\leq i\leq t\end{subarray}}(\psi_{g_{1}}\psi_{g_{2}}\psi_{h_{1}}\cdots\psi_{h_{t}})(\alpha).

(6)

Thus, we define the characteristic function $\mathcal{Q}$ for the set $S_{k}$ of all $k$ -normal elements as $\mathcal{Q}=\sum_{g\in P_{k}}\mathcal{Q}_{g}^{G}$ .

Now, for a divisor $H$ of $G$ , we define a collection $T_{g,k}^{H}$ of elements $\alpha$ of $\mathbb{F}_{q^{n}}$ such that $\alpha$ is $H$ -free, $\alpha=g\circ\beta$ for some $\beta\in\mathbb{F}_{q^{n}}$ , but $\alpha\neq\Lambda_{i}\circ\beta$ for any $\beta\in\mathbb{F}_{q^{n}}$ and for any $i=1,2,\ldots,t.$ We define the characteristic function $\mathcal{Q}_{g}^{H}$ for the set $T_{g,k}^{H}$ similar to that for the set $S_{g,k}$ as follows

\displaystyle\mathcal{Q}_{g}^{H}(\alpha)=

\displaystyle\frac{\Phi_{q}(H)}{q^{\mathrm{deg}(H)+\mathrm{deg}(g)}}\sum_{\begin{subarray}{c}g_{1}\mid H,\\ g_{2}\mid\pi\end{subarray}}\sum_{\begin{subarray}{c}h_{i}\mid\Lambda_{i},\\ 1\leq i\leq t\end{subarray}}\frac{\mu^{\prime}(g_{1})}{\Phi_{q}(g_{1})}\big{(}\prod_{i=1}^{t}\ell_{f_{i},h_{i}}^{\prime}\big{)}\sum_{\begin{subarray}{c}\psi_{g_{1}},\psi_{g_{2}},\\ \psi_{h_{i}};\ 1\leq i\leq t\end{subarray}}(\psi_{g_{1}}\psi_{g_{2}}\psi_{h_{1}}\cdots\psi_{h_{t}})(\alpha).

(7)

Observe that, in particular, if $H=G$ , then $T_{g,k}^{H}=S_{g,k}$ and $\mathcal{Q}_{g}^{H}=\mathcal{Q}_{g}^{G}.$

Henceforth, the notation used in this section shall have the same meaning throughout the article unless otherwise stated.

4 Existence of an $r$ -primitive $k$ -normal pair $(\alpha,\alpha^{-1})$

In this section, our aim is to show the existence of a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}.$ From Lemma 2.1, it is clear that, if $\gamma\in\mathbb{F}_{q^{n}}$ is a normal element and $g\in P_{k}$ , then $\alpha=g\circ\gamma$ is a $k$ -normal element over $\mathbb{F}_{q}.$ Therefore, for the existence of a desired pair, we only need to check the $r$ -primitivity of $\alpha$ and the $k$ -normality of $\alpha^{-1}$ . This means for the existence of a desired pair, it is enough to find an $(x^{n}-1)$ -free element $\gamma$ , such that $\alpha=g\circ\gamma\in Q_{r}^{R}$ and $\alpha^{-1}\in T_{g,k}^{G}$ , for some arbitrary but fixed polynomial $g\in P_{k}.$ Instead of this, more generally, we will show the existence of an $h$ -free element $\gamma$ , for some $h\mid x^{n}-1$ , such that $\alpha=g\circ\gamma\in Q_{r}^{d}$ and $\alpha^{-1}\in T_{g,k}^{H}$ , where $d\mid R$ and $H\mid G$ . Let $N_{r,k,g}(h,d,H)$ denote the number of such elements $\gamma$ , then we prove the following result.

Theorem 4.1.

Let $r>0$ and $k\geq 0$ be integers. Further, let $q$ be a prime power and $n$ be a positive integer such that $r\mid q^{n}-1$ and there exists a polynomial $g\in P_{k}$ . Then, $N_{r,k,g}(h,d,H)>0$ , if

q^{n/2-\vartheta}>2r\hskip 1.42262pt\mathrm{rad}(r)W(h)W(d)W(H),

where, $\vartheta=2k$ , if $\mathrm{gcd}(q,n)=1$ , and $\vartheta=3k$ , otherwise.

Proof.

Let $g=\sum_{i=0}^{k}a_{i}x^{i}\in P_{k}$ with $a_{k}=1$ , and define $g\circ x=\sum_{i=0}^{k}a_{i}x^{q^{i}}$ . Further, let $\mathcal{Z}$ be the set of zeroes of the polynomial $g\circ x$ . Clearly, $\mathcal{Z}$ consists of all the elements $\beta$ , whose $\mathbb{F}_{q}$ -order divides $g$ , and hence, $|\mathcal{Z}|=\sum_{g^{\prime}\mid g}\Phi_{q}(g^{\prime})=q^{k}$ . Now, by using the definitions of the characteristic functions $\Upsilon_{h}$ , $\Gamma_{r}^{d}$ and $\mathcal{Q}_{g}^{H}$ , we have

$\displaystyle N_{r,k,g}(h,d,H)=$	$\displaystyle\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\Upsilon_{h}(\beta)\Gamma_{r}^{d}(g\circ\beta)\mathcal{Q}_{g}^{H}((g\circ\beta)^{-1})$	(8)
$\displaystyle=$	$\displaystyle\hskip 2.84526pt\mathcal{H}\sum_{\begin{subarray}{c}h^{\prime}\mid h\end{subarray}}\sum_{\begin{subarray}{c}d_{1}\mid d,d_{2}\mid u,\\ e_{j}\mid\lambda_{j};1\leq j\leq s\end{subarray}}\sum_{\begin{subarray}{c}g_{1}\mid H,g_{2}\mid\pi,\\ h_{i},\mid\Lambda_{i};1\leq i\leq t\end{subarray}}\frac{\bm{\mu}}{\bm{\Phi}}(h^{\prime},d_{1},g_{1})\bm{\ell}\big{(}(e_{j})_{j=1}^{s},(h_{i})_{i=1}^{t}\big{)}\sum_{\begin{subarray}{c}\psi_{h^{\prime}}\end{subarray}}\times$
	$\displaystyle\sum_{\begin{subarray}{c}\chi_{d_{1}},\chi_{d_{2}},\\ \chi_{e_{j}};1\leq j\leq s\end{subarray}}\sum_{\begin{subarray}{c}\psi_{g_{1}},\psi_{g_{2}},\\ \psi_{h_{i}};\ 1\leq i\leq t\end{subarray}}\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\psi_{h^{\prime}}(\beta)\chi_{D}(g\circ\beta)\psi_{F}((g\circ\beta)^{-1}),$

where $\mathcal{H}=\dfrac{\Phi_{q}(h)\phi(d)\Phi_{q}(H)}{rdq^{\mathrm{deg}(h)+\mathrm{deg}(H)+\mathrm{deg}(g)}}$ , $\dfrac{\bm{\mu}}{\bm{\Phi}}(h^{\prime},d_{1},g_{1})=\dfrac{\mu^{\prime}(h^{\prime})\mu(d_{1})\mu^{\prime}(g_{1})}{\Phi_{q}(h^{\prime})\phi(d_{1})\Phi_{q}(g_{1})}$ , $\bm{\ell}\big{(}(e_{j})_{j=1}^{s},(h_{i})_{i=1}^{t}\big{)}=\prod_{j=1}^{s}\ell_{p_{j},e_{j}}\prod_{i=1}^{t}\ell_{f_{i},h_{i}}^{\prime}$ , $\chi_{D}=\chi_{d_{1}}\chi_{d_{2}}\chi_{e_{1}}\cdots\chi_{e_{s}}$ is a multiplicative character of order $D=d_{1}d_{2}e_{1}\cdots e_{s}$ , since $d_{1},d_{2},e_{1},\ldots,e_{s}$ are mutually co-prime, and similarly, $\psi_{F}=\psi_{g_{1}}\psi_{g_{2}}\psi_{h_{1}}\cdots\psi_{h_{t}}$ is the additive character of $\mathbb{F}_{q}$ -order $F=g_{1}g_{2}h_{1}\cdots h_{t}$ .

Now, let $\psi_{0}$ be the canonical additive character of $\mathbb{F}_{q^{n}}$ , then there exist $y_{1},y_{2}\in\mathbb{F}_{q^{n}}$ , such that $\psi_{h^{\prime}}(\beta)=\psi_{0}(y_{1}\beta)$ and $\psi_{F}(g\circ\beta)=\psi_{0}(y_{2}(g\circ\beta))$ for all $\beta\in\mathbb{F}_{q^{n}}$ . Thus,

\displaystyle\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\psi_{h^{\prime}}(\beta)\chi_{D}(g\circ\beta)\psi_{F}((g\circ\beta)^{-1})=\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta)\psi_{0}(y_{1}\beta+y_{2}(g\circ\beta)^{-1}).

Now, we estimate $\big{|}\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta)\psi_{0}(y_{1}\beta+y_{2}(g\circ\beta)^{-1})\big{|}$ using Lemma 2.4, for which we first show that the rational function $y_{1}x+y_{2}(g\circ x)^{-1}$ is not of the form $L(x)^{q^{n}}-L(x)$ for any rational function $L(x)=\frac{L_{1}}{L_{2}}$ (in the simplest form) over the algebraic closure $\mathbb{F}$ of $\mathbb{F}_{q^{n}}$ , unless $y_{1}=y_{2}=0$ .

Assume that, $y_{1}x+y_{2}(g\circ x)^{-1}=(\frac{L_{1}}{L_{2}})^{q^{n}}-\frac{L_{1}}{L_{2}}$ for some $L_{1},L_{2}\in\mathbb{F}[x]$ i.e. $L_{2}^{q^{n}}(y_{1}\sum_{i=0}^{k}a_{i}x^{q^{i}+1}+y_{2})=\sum_{i=0}^{k}a_{i}x^{q^{i}}(L_{1}^{q^{n}}-L_{1}L_{2}^{q^{n}-1}).$ First, let $y_{2}\neq 0$ , then $\sum_{i=0}^{k}a_{i}x^{q^{i}}=L_{2}^{q^{n}},$ which is not possible because $k<n$ . So, $y_{2}$ must be zero. Additionally, let $y_{1}\neq 0$ , then $L_{2}^{q^{n}}y_{1}x=L_{1}^{q^{n}}-L_{1}L_{2}^{q^{n}-1}$ , which implies $L_{2}$ is a constant polynomial and $q^{n}\mid 1$ , which is again not possible. Hence, $y_{1}=y_{2}=0$ . Therefore, if one of the $y_{1}$ or $y_{2}$ is non-zero, then by Lemma 2.4,

\big{|}\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta)\psi_{0}(y_{1}\beta+y_{2}(g\circ\beta)^{-1})\big{|}\leq 2q^{n/2+k}.

(I)

Now, let $y_{1}=y_{2}=0$ , then $\psi_{h^{\prime}}$ and $\psi_{F}$ are trivial additive characters, i.e. $h^{\prime}=1=F$ , and the latter one is possible if and only if each factor of $F=g_{1}g_{2}h_{1}\cdots h_{t}$ is $1$ , since $g_{1},g_{2},h_{1},\ldots,h_{t}$ are mutually co-prime. Hence, we have

\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta)\psi_{0}(y_{1}\beta+y_{2}(g\circ\beta)^{-1})=\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta).

Notice that, if the polynomial $g\circ x=L(x)^{D}$ for some $L(x)\in\mathbb{F}[x]$ , then $D\mid\mathrm{gcd}(q^{k},q^{n}-1)$ , i.e $D=1$ . Hence, by Lemma 2.3, for $D\neq 1$ , we get

\big{|}\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta)\big{|}\leq(q^{k}-1)q^{n/2}<2q^{n/2+k}.

(II)

Finally, let $h^{\prime}=F=1$ and $D=1$ , then

\sum_{\beta\in\mathbb{F}_{q^{n}}\setminus\mathcal{Z}}\chi_{D}(g\circ\beta)\psi_{0}(y_{1}\beta+y_{2}(g\circ\beta)^{-1})=q^{n}-q^{k}.

(III)

Using the estimates, obtained in (I), (II) and (III), in Equation (8), we get

	$\displaystyle\big{\|}N_{r,k,g}(h,d,H)-$	$\displaystyle(q^{n}-q^{k})\bm{\ell}\big{(}(1)_{j=1}^{s},(1)_{i=1}^{t}\big{)}\mathcal{H}\big{\|}\leq$
		$\displaystyle\mathcal{H}\underbrace{\sum_{\begin{subarray}{c}h^{\prime}\mid h\end{subarray}}\sum_{\begin{subarray}{c}d_{1}\mid d,d_{2}\mid u,\\ e_{j}\mid\lambda_{j};1\leq j\leq s\end{subarray}}\sum_{\begin{subarray}{c}g_{1}\mid H,g_{2}\mid\pi,\\ h_{i},\mid\Lambda_{i};1\leq i\leq t\end{subarray}}}_{\text{(all are not 1 simultaneously)}}\big{\|}\frac{\bm{\mu}}{\bm{\Phi}}(h^{\prime},d_{1},g_{1})\big{\|}\big{\|}\bm{\ell}\big{(}(e_{j})_{j=1}^{s},(h_{i})_{i=1}^{t}\big{)}\big{\|}\times$
		$\displaystyle\sum_{\begin{subarray}{c}\psi_{h}^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}\chi_{d_{1}},\chi_{d_{2}},\\ \chi_{e_{j}};1\leq j\leq s\end{subarray}}\sum_{\begin{subarray}{c}\psi_{g_{1}},\psi_{g_{2}},\\ \psi_{h_{i}};\ 1\leq i\leq t\end{subarray}}2q^{n/2+k}.$

Since $\big{|}\bm{\ell}\big{(}(e_{j})_{j=1}^{s},(h_{i})_{i=1}^{t}\big{)}\big{|}\leq\bm{\ell}\big{(}(1)_{j=1}^{s},(1)_{i=1}^{t}\big{)}$ , then

	$\displaystyle\big{\|}$	$\displaystyle N_{r,k,g}(h,d,H)-(q^{n}-q^{k})\bm{\ell}\big{(}(1)_{j=1}^{s},(1)_{i=1}^{t}\big{)}\mathcal{H}\big{\|}\leq$
		$\displaystyle\hskip 1.42262pt2q^{n/2+k}\bm{\ell}\big{(}(1)_{j=1}^{s},(1)_{i=1}^{t}\big{)}\mathcal{H}\big{(}u\prod_{j=1}^{s}\lambda_{j}q^{\mathrm{deg}(\pi)+\sum_{i=1}^{t}\mathrm{deg}(\Lambda_{i})}W(h)W(d)W(H)-1\big{)}.$

From the above equation, we get

	$\displaystyle\frac{N_{r,k,g}(h,d,H)}{\bm{\ell}\big{(}(1)_{j=1}^{s},(1)_{i=1}^{t}\big{)}\mathcal{H}}$	$\displaystyle\geq(q^{n}-q^{k})-2q^{n/2+k}\times$
		$\displaystyle\big{(}u\prod_{j=1}^{s}\lambda_{j}q^{\mathrm{deg}(\pi)+\sum_{i=1}^{t}\mathrm{deg}(\Lambda_{i})}W(h)W(d)W(H)-1\big{)}.$

Clearly, $N_{r,k,g}(h,d,H)>0$ , if

q^{n/2-k}>2u\prod\limits_{j=1}^{s}\lambda_{j}q^{\mathrm{deg}(\pi)+\sum_{i=1}^{t}\mathrm{deg}(\Lambda_{i})}W(h)W(d)W(H).

(9)

Now, if $\mathrm{gcd}(q,n)=1$ , then $x^{n}-1$ has no repeated factor over $\mathbb{F}_{q}$ , and hence, $g=\pi$ and $\mathrm{deg}(\Lambda_{i})=0$ for each $i=1,2,\ldots,t.$ Thus, $q^{\mathrm{deg}(\pi)+\sum_{i=1}^{t}\mathrm{deg}(\Lambda_{i})}=q^{k}$ , otherwise $\leq q^{2k}.$ Moreover, $u\prod\limits_{j=1}^{s}\lambda_{j}=r\prod_{j=1}^{s}p_{j}\leq r\hskip 1.42262pt\mathrm{rad}(r)$ . Hence, from Equation (9), $N_{r,k,g}(h,d,H)>0$ , if $q^{n/2-\vartheta}>2r\hskip 1.42262pt\mathrm{rad}(r)W(h)W(d)W(H)$ , where, $\vartheta=2k$ , if $\mathrm{gcd}(q,n)=1$ , and $\vartheta=3k$ , otherwise. ∎

The following corollary provides the desired existence of $r$ -primitive $k$ -normal pairs $(\alpha,\alpha^{-1})$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}.$

Corollary 4.1.1.

With $r$ and $k$ as in Theorem 4.1, there exists a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , i.e. $N_{r,k,g}(x^{n}-1,R,G)>0$ , if

q^{n/2-\vartheta}>2r\hskip 1.42262pt\mathrm{rad}(r)W(x^{n}-1)W(R)W(G),

(10)

where, $\vartheta=2k$ , if $\mathrm{gcd}(q,n)=1$ , and $\vartheta=3k$ , otherwise.

Proof.

Let $h=x^{n}-1$ , $d=R$ and $H=G$ in the above theorem, then the result follows immediately. ∎

Observe that, Inequalities (10) can never hold for $n\leq 2\vartheta$ , and as the value of $k$ increases, the existence of a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ becomes more challenging for all $n.$ However, we can show that such a pair exists in all but finitely many fields $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , for any fixed integers $r$ and $k$ . For this, we need the following two lemmas.

Lemma 4.1.

[10, Lemma 2.9] Let $q$ be a prime power and $n$ be a positive integer. Then, we have $W(x^{n}-1)\leq 2^{\frac{1}{2}(n+\mathrm{gcd}(n,q-1))}$ . In particular, $W(x^{n}-1)\leq 2^{n}$ and $W(x^{n}-1)=2^{n}$ if and only if $n\mid(q-1)$ . Furthermore, $W(x^{n}-1)\leq 2^{3n/4}$ if $n\nmid(q-1)$ , since in this case, $\mathrm{gcd}(n,q-1)\leq n/2.$

Lemma 4.2.

[3, Lemma 3.7] For any $M\in\mathbb{N}$ and a positive real number $\nu$ , $W(M)\leq\mathcal{C}_{\nu}M^{1/\nu}$ , where $\mathcal{C}_{\nu}=\prod\limits_{i=1}^{t}\frac{2}{p_{i}^{1/\nu}}$ and $p_{1},p_{2},\ldots,p_{t}$ are the primes $\leq 2^{\nu}$ that divide $M.$

Following the idea of [18, Proposition 3.1], we have the following result.

Proposition 4.1.

Let $r\geq 1$ and $k\geq 0$ be fixed integers. Let $q$ be a prime power and $n>2\vartheta$ , where $\vartheta$ is as defined in Theorem 4.1, such that $r\mid q^{n}-1$ and there exists a polynomial $g\in P_{k}$ . Then, there exists a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in all but finitely many fields $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}.$

Proof.

Notice that, $W(R)\leq W(q^{n}-1)$ and $W(G)\leq W(x^{n}-1).$ Hence, from Lemmas 4.1 and 4.2, Inequality (10) holds if

q^{n/2-\vartheta}>2r\hskip 1.42262pt\mathrm{rad}(r)\mathcal{C}_{\nu}q^{n/\nu}2^{2n}.

(11)

Set $A=2r\hskip 1.42262pt\mathrm{rad}(r)\mathcal{C}_{\nu}$ and $\nu=\mathrm{max}\hskip 1.42262pt\{4\vartheta+2,93.46\}.$ Then, taking logarithm on both sides, we get the following equivalent inequality.

\mathrm{log}\hskip 1.42262ptq>\frac{\mathrm{log}\hskip 1.42262ptA+2n\mathrm{log}\hskip 1.42262pt2}{n/2-\vartheta-n/\nu}.

The right hand side of the above inequality is a decreasing function of $n$ , if $\frac{n}{2}-\vartheta-\frac{n}{\nu}>0$ , i.e. $\frac{\nu-2}{\nu}>\frac{2\vartheta}{n}$ , which holds for $n>2\vartheta$ . Therefore, there exists a number $q_{0}$ such that the above inequality holds for $q\geq q_{0}$ and $n\geq 2\vartheta+1$ .

Now, for $q<q_{0}$ , we consider the following inequality, which is equivalent to Inequality (11).

n>\frac{\mathrm{log}\hskip 1.42262ptA+\vartheta\hskip 1.42262pt\mathrm{log}\hskip 1.42262ptq}{(1/{2}-1/{\nu})\mathrm{log}\hskip 1.42262ptq-2\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2}.

Clearly, the denominator of the right hand side of the above inequality is positive, if $q>16$ . Hence, corresponding to each $16<q<q_{0}$ , there exists a natural number $n_{q}$ such that the above inequality holds for $n\geq n_{q}.$

Now, for $3\leq q\leq 16$ , we use the bound $W(x^{n}-1)<2^{n/3+c_{q}}$ (from [10, Lemmas 2.9, 2.11]), where $c_{q}$ is a constant corresponding to each $q$ . Then, Inequality (11) holds, if

n>\frac{\mathrm{log}\hskip 1.42262ptA+\vartheta\hskip 1.42262pt\mathrm{log}\hskip 1.42262ptq+2c_{q}\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2}{(1/{2}-1/{\nu})\mathrm{log}\hskip 1.42262ptq-(2\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2)/{3}}.

The above inequality holds for $n\geq n_{q}$ , where $n_{q}$ is a natural number corresponding to each $3\leq q\leq 16$ .

Finally, let $q=2$ , and from [10, Lemma 2.11], we have $W(x^{n}-1)<2^{\frac{n-4}{5}}$ for all $n$ except $n\in\{1,2,3,4,5,7,9,15,21\}$ , and Inequality (11) holds, if

n>\frac{\mathrm{log}\hskip 1.42262ptA+\vartheta\hskip 1.42262pt\mathrm{log}\hskip 1.42262ptq-(8/5)\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2}{(1/{2}-1/{\nu})\mathrm{log}\hskip 1.42262ptq-(2\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2)/{5}}.

There exists a natural number $n_{2}$ such that the last inequality holds for $n\geq n_{2}.$

Hence, from the above discussion, we conclude that, there exists a pair $(\alpha,\alpha^{-1})$ of $r$ -primitive $k$ -normal elements in all but finitely many $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}.$ ∎

4.1 The sieving technique

In this section, we give a sieving technique to improve Inequality (10) by following the methods of Cohen and Huczynska [3]. The proofs of the next two lemmas are similar to that of [14, Lemmas 5.1, 5.2], hence omitted.

Lemma 4.3.

Let $r\mid q^{n}-1$ , and let $k$ be a non-negative integer and $g\in P_{k}$ . Further, let $d$ be a divisor of $R$ and $\{p_{1},p_{2},\ldots,p_{l}\}$ be the set of remaining distinct primes dividing $R$ . Furthermore, let $h$ , $H$ be the divisors of $x^{n}-1$ , $G$ respectively, and $\{h_{1},h_{2},\ldots,h_{m}\}$ , $\{H_{1},H_{2},\ldots,H_{m^{\prime}}\}$ be the sets of remaining distinct irreducible factors of $x^{n}-1$ , $G$ respectively. Then

	$\displaystyle N_{r,k,g}(x^{n}-1,R,G)\geq$	$\displaystyle\sum_{i=1}^{m}N_{r,k,g}(hh_{i},d,H)+\sum_{i=1}^{l}N_{r,k,g}(h,dp_{i},H)+$
		$\displaystyle\sum_{i=1}^{m^{\prime}}N_{r,k,g}(h,d,HH_{i})-(m+l+m^{\prime}-1)N_{r,k,g}(h,d,H).$

Lemma 4.4.

With the notations of the above lemma, define

\mathcal{D}:=1-\sum_{i=1}^{m}\frac{1}{q^{deg(h_{i})}}-\sum_{i=1}^{l}\frac{1}{p_{i}}-\sum_{i=1}^{m^{\prime}}\frac{1}{q^{deg(H_{i})}}\text{ and }\mathcal{S}:=\frac{m+l+m^{\prime}-1}{\mathcal{D}}+2.

Suppose $\mathcal{D}>0,$ then $N_{r,k,g}(x^{n}-1,R,G)>0$ , if

q^{n/2-\vartheta}>2r\hskip 1.42262pt\mathrm{rad}(r)W(h)W(d)W(H)\mathcal{S},

(12)

where, $\vartheta=2k$ , if $\mathrm{gcd}(q,n)=1$ , and $\vartheta=3k$ , otherwise.

5 Existence of a primitive $1$ -normal pair $(\alpha,\alpha^{-1})$

In [3], Cohen and Huczynska proved the existence of a pair $(\alpha,\alpha^{-1})$ of primitive normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ . Here, we present an application of the sufficient condition and the sieving inequality, obtained in the previous section, by showing the existence of a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ . All the non-trivial calculations wherever needed in this section are done using SageMath [19].

From Inequality (10), there exist pairs $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , if

q^{n/2-\vartheta}>2W(x^{n}-1)W(R)W(G),

(13)

where $\vartheta=2$ , if $\mathrm{gcd}(q,n)=1$ , otherwise, $\vartheta=3.$ Notice that, if a pair $(q,n)$ satisfies Inequality (13) in the case of $\vartheta=3$ , then it satisfies the same in the case of $\vartheta=2$ as well. Moreover, Inequality (13) can never hold for $n\leq 4$ , if $\vartheta=2$ , and for $n\leq 6$ , if $\vartheta=3.$ Therefore, first we assume that $\vartheta=3$ and settle the cases $n\geq 14$ in Subsection 5.1, and then discuss the cases $5\leq n\leq 13$ in Subsection 5.2.

5.1 The cases $n\geq 14$

From Lemmas 4.1 and 4.2, we have $W(R)\leq W({q^{n}-1})<\mathcal{C}_{\nu}q^{n/\nu}$ , $W(x^{n}-1)\leq 2^{n}$ and $W(G)\leq 2^{n-1}$ , and hence, Inequality (13) holds, if

q^{n/2-3}>\mathcal{C}_{\nu}\hskip 1.42262ptq^{n/\nu}2^{2n}.

(14)

First, we study the cases $q>3$ and $n\geq 14$ in the following Lemma and then the cases $q=2,3$ and $n\geq 14$ in Lemma 5.3.

Lemma 5.1.

Let $q>3$ be a prime power and $n\geq 14$ be a positive integer. Then, there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ .

Proof.

For $n\geq 14$ , if we choose $\nu=7.6$ , Inequality (14) holds for $q\geq 214183$ . Now, for $q<214183$ , we rewrite Inequality (14) as follows

n>\frac{\mathrm{log}\hskip 1.42262pt(\mathcal{C}_{\nu})+3\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt(q)}{(1/{2}-1/{\nu})\mathrm{log}\hskip 1.42262ptq-2\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2},

(15)

provided $\nu>\frac{2\hskip 1.42262pt\mathrm{log}\hskip 1.42262ptq}{\mathrm{log}\hskip 1.42262ptq-4\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2}$ and $q>16.$ Notice that, the value of $\nu$ is large for small values of $q$ , and it is difficult to calculate $\mathcal{C}_{\nu}$ in our computer. So, for simplicity, we assume $37\leq q<214183,$ and for each such $q$ , we find a natural number $n_{q}$ (see Table 1) such that Inequality (15) holds for $n\geq n_{q}$ .

Table 1: Values of

n_{q}

for

37\leq q<214183

using

W(x^{n}-1)\leq 2^{n}

$\nu$	$q$	$n_{q}$	$\nu$	$q$	$n_{q}$	$\nu$	$q$	$n_{q}$
7.6	$\geq 214183$	14	7.8	307, 311, 313, 317	34	8.3	107	62
7.5	61747 to 214182	15	7.8	277, 281, 283, 289, 293	35	8.3	103	64
7.5	24428 to 61746	16	7.8	257, 263, 269, 271	36	8.3	101	65
7.5	11926 to 24427	17	7.8	243, 251, 256	37	8.4	97	67
7.5	6738 to 11925	18	7.8	229, 233, 239, 241	38	8.5	89	73
7.5	4231 to 6737	19	7.9	223, 227	39	8.5	83	79
7.5	2875 to 4230	20	7.9	211	40	8.6	81	81
7.5	2074 to 2874	21	7.9	197, 199	41	8.6	79	83
7.5	1569 to 2073	22	7.9	191, 193	42	8.7	73	92
7.5	1233 to 1568	23	7.9	179, 181	43	8.8	71	96
7.5	999 to 1232	24	7.9	173	44	8.8	67	104
7.5	829 to 998	25	8	167, 169	45	8.9	64	111
7.5	703 to 828	26	8	157, 163	47	9	61	120
7.6	606 to 702	27	8	149, 151	48	9.1	59	128
7.6	531 to 605	28	8.1	137, 139	51	9.3	53	157
7.6	471 to 530	29	8.1	128, 131	53	9.6	49	188
7.6	422 to 470	30	8.1	125, 127	54	9.7	47	210
7.6	382 to 421	31	8.1	121	56	10.1	43	272
7.7	348 to 381	32	8.2	113	59	10.3	41	320
7.7	331, 337, 343, 347	33	8.3	109	61	10.8	37	489

Now, for $3<q<37$ , we use the bound $W(x^{n}-1)\leq 2^{n/3+2(q^{2}-1)/3}$ , which we obtain from [10, Lemma 2.9], and find that, Inequality (14) holds, if

n>\frac{\mathrm{log}\hskip 1.42262pt(2^{4(q^{2}-1)/3}\mathcal{C}_{\nu})}{(1/{2}-1/{\nu})\mathrm{log}\hskip 1.42262ptq-(2\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2)/3},

(16)

provided $\nu>\frac{6\hskip 1.42262pt\mathrm{log}\hskip 1.42262ptq}{3\hskip 1.42262pt\mathrm{log}\hskip 1.42262ptq-4\hskip 1.42262pt\mathrm{log}\hskip 1.42262pt2}$ and $q>2$ . For these values of $q$ , we obtain a natural number $n_{q}$ (see Table 2) such that Inequality (16) holds for $n\geq n_{q}$ .

Now, for $n<n_{q}$ and $q$ ’s listed in Tables 1 and 2, we test Inequality (13) with $\vartheta=3$ , and list the values of $q$ and $n$ in Table 3, which fail to satisfy this inequality.

Table 2: Values of

n_{q}

for

3<q<37

using

W(x^{n}-1)\leq 2^{n/3+2(q^{2}-1)/3}

$\nu$	11.8	11.7	11.6	11.4	11.2	11	10.6	10.3
$q$	32	31	29	27	25	23	19	17
$n_{q}$	1019	973	883	797	714	636	493	428
$\nu$	10.1	9.7	9.4	9	8.8	8.7	8.5	9
$q$	16	13	11	9	8	7	5	4
$n_{q}$	397	312	263	222	206	195	214	351

Table 3: Pairs

(q,n)

fail to satisfy Inequality (13) with

\vartheta=3

$n$	$q$
14	4, 5, 8, 9, 11, 13, 23, 25, 27, 29, 41, 43, 64, 71, 113, 125, 127, 169, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 729, 743, 883, 911, 953
15	4, 7, 8, 11, 13, 16, 19, 29, 31, 41, 49, 61, 64, 71, 121, 151, 181, 211, 241, 256, 271, 331, 361, 421, 571, 631, 751, 841
16	5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 41, 49, 73, 81, 89, 97, 113, 193, 241, 257, 289, 337, 401
17	4, 16, 103, 256
18	4, 5, 7, 13, 17, 19, 25, 31, 37, 43, 73, 109, 127, 163, 181, 199, 289, 361
20	7, 9, 11, 13, 19, 29, 31, 41, 61, 81, 101, 121
21	4, 8, 13, 16, 43, 64, 169
22	23, 67, 89
23	47
24	5, 7, 11, 13, 17, 19, 25, 37, 49, 73, 97, 121
26	27, 53
28	13, 29
30	4, 7, 11, 19, 31, 61
31	32
32	17
36	5, 19, 37
40	9, 11, 41
42	43
45	4
48	5, 7
63	5

Now, for the values of $q$ and $n$ , listed in Table 3, we test the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=3$ , in which we choose the suitable values of $h$ , $d$ and $H$ , and test Inequality (12) (whose pseudocode is given in A), which returns True for all the pairs $(q,\ n)$ except (4, 15), (5, 16), (5, 24), (8, 14), (9, 16), (16, 15), (17, 16), (19, 18). For these remaining pairs of $(q,\ n)$ , we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n) (pseudocode is given in A), and find that such a pair always exists. ∎

Now, for the cases $q=2,\ 3$ , first we recall some results. Let $q$ be a prime power and $n=q^{i}\cdot n^{\prime}$ be a positive integer with $\mathrm{gcd}(n^{\prime},q)=1$ ; $i\geq 0$ . Further, let $e$ be the multiplicative order of $q$ modulo $n^{\prime}$ , then $x^{n^{\prime}}-1$ factors into irreducible polynomials of degree $\leq e.$ Let $\rho(q,n^{\prime})$ be the ratio of the number of irreducible factors of $x^{n^{\prime}}-1$ of degree $<e$ to $n^{\prime}$ , then $n\rho(q,n)=n^{\prime}\rho(q,n^{\prime})$ , and we have the following lemma.

Lemma 5.2.

[3, Lemma 7.1] Assume that $n>4$ $(p\nmid n)$ , then the following holds:

$(i)$

Suppose $q=2$ , then $\rho(2,5)=1/5;\ \rho(2,9)=2/9;\ \rho(2,21)=4/21;$ otherwise $\rho(2,n)\leq 1/6.$
$(ii)$

Suppose $q=3$ , then $\rho(3,16)=5/16;$ otherwise $\rho(3,n)\leq 1/4.$

If we choose $h$ as the product of all irreducible factors of $x^{n^{\prime}}-1$ of degree $<e$ , $H=h/(x-1)$ and $d=R$ in Lemma 4.4, then following Lemma 10 of [15], we get that, $\mathcal{D}>1-2/e>0$ and $\mathcal{S}\leq 2n^{\prime}\leq 2n$ , provided $e>2$ (i.e. $n^{\prime}\nmid q^{2}-1$ ). Using these results, we prove the following lemma.

Lemma 5.3.

Let $q=2,3$ and $n\geq 14$ be a positive integer. Then, there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ .

Proof.

Let $n=q^{i}\cdot n^{\prime}$ with $\mathrm{gcd}(q,n^{\prime})=1$ and $i\geq 0.$ First, assume that $n^{\prime}\nmid q^{2}-1,$ then $e>2$ and $n^{\prime}>4$ . If we choose $d=R$ , $h$ as the product of all irreducible factors of $x^{n^{\prime}}-1$ of degree $<e$ , and $H=h/(x-1)$ , then from the above discussion and Inequality (12), there exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , if

q^{n/2-3}>2\mathcal{C}_{\nu}q^{n/\nu}W(h)W(H)\mathcal{S}.

(17)

Notice that, $W(H)=W(h)/2$ and $W(h)=2^{n^{\prime}\rho(q,n^{\prime})}$ . From Lemma 5.2 $n^{\prime}\rho(q,n^{\prime})\leq n^{\prime}/6\leq n/6,$ when $q=2$ , and $n^{\prime}\rho(q,n^{\prime})\leq n^{\prime}/4\leq n/4,$ when $q=3$ . Thus, the above inequality holds, if

\left\{\begin{array}[]{ll}q^{n/2-3}>\mathcal{C}_{\nu}q^{n/\nu}2^{n/3}2n,&\text{ for }q=2,\\ q^{n/2-3}>\mathcal{C}_{\nu}q^{n/\nu}2^{n/2}2n,&\text{ for }q=3.\end{array}\right.

The above inequalities hold for $n\geq 557$ , when $q=2$ and $\nu=8.7$ , and for $n\geq 265$ , when $q=3$ and $\nu=8.3.$ For the remaining $q$ and $n$ , we test Inequality (13), and find that this fails for $n=$ 14, 15, 17, 18, 20, 21, 22, 27, 28, 30, 31, 33, 35, 42, 45, 63, when $q=2$ , and for $n=$ 14, 15, 16, 20, 22, 26, 28, 32, 40, when $q=3.$

Now, assume that $n^{\prime}\mid q^{2}-1$ , and select $d=R$ , $h$ as the product of all the linear factors of $x^{n^{\prime}}-1$ , and $H=h/(x-1)$ in Lemma 4.4. Then, for $q=2$ , there exists exactly one linear and at most 1 quadratic factor of $x^{n^{\prime}}-1$ over $\mathbb{F}_{2}$ . Therefore, in this case, $\mathcal{D}\geq 1/2$ and $\mathcal{S}\leq 4$ . Similarly, for $q=3$ , there exists at most 2 linear and at most 2 quadratic factors of $x^{n^{\prime}}-1$ over $\mathbb{F}_{3}$ . Thus, in this case, $\mathcal{D}\geq 1/3$ and $\mathcal{S}\leq 17$ . If we take $\nu=6$ , then Inequality (17) holds for $n\geq 33$ in both the cases. For $14\leq n<33$ , we test Inequality (13), which does not hold for $n=$ 16, when $q=2$ , and for $n=$ 18, 24, when $q=3.$

Finally, for $q=2,3$ and the remaining values of $n$ , we apply the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=3$ , which returns False only for $q=2$ and $n=$ 14, 15, 16, 18, 20, 21, 24, 30, and for $q=3$ and $n=$ 14, 16. For these remaining $q$ and $n$ , we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists. ∎

5.2 The cases $5\leq n\leq 13$

Observe that, Inequality (13) is applicable for the cases, $n=5$ , 6, only if $\mathrm{gcd}(n,q)=1$ . Therefore, first we settle the cases $7\leq n\leq 13$ for all $q\geq 2$ , and then the cases $n=5$ , 6 with the assumption that $\mathrm{gcd}(q,n)=1.$ Before moving ahead, first we state the following lemma, which is a direct consequence of Lemma 4.4.

Lemma 5.4.

Let $d$ be a divisor of $q^{n}-1$ and $n_{0}$ be a fixed positive integer such that all the prime divisors of $(q^{n}-1)/d$ , co-prime to $d$ , are of the form $n_{0}s+1$ for some positive integer $s$ . Further, let $p_{1},p_{2},\ldots,p_{l}$ be the first $l$ primes of the form $n_{0}s+1$ such that the product $P_{l}=\prod_{i=1}^{l}p_{i}\leq(q^{n}-1)/d$ , and denote the sum of their reciprocals by $S_{l}=\sum_{i=1}^{l}{1}/{p_{i}}$ . If $h=1$ and $H=1$ are the divisors of $x^{n}-1$ and $(x^{n}-1)/(x-1)$ , respectively, then $\mathcal{D}\geq 1-S_{l}-(2n-1)/q$ and $\mathcal{S}\leq(l+2n-2)/\mathcal{D}+2.$ Moreover, if $\mathcal{D}>0$ and $q^{n/2-\vartheta}>2W(d)\mathcal{S}$ , then there exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , where $\mathcal{D}$ , $\mathcal{S}$ and $\vartheta$ are the same as defined in Lemma 4.4 and Inequality (13).

Lemma 5.5.

Let $7\leq n\leq 13$ and $q$ be a prime power such that $\mathrm{gcd}(q,n)=1$ . Then, there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ .

Proof.

Since, $\mathrm{gcd}(q,n)=1$ , Inequality (13) holds, if

q^{n/2-2}>\mathcal{C}_{\nu}q^{n/\nu}2^{2n}.

(18)

For each $7\leq n\leq 13$ and suitable choices of $\nu$ , the above inequality holds for $q\geq M_{n}$ , where the values of $M_{n}$ are listed in Table 4.

Table 4: Values of

M_{n}

such that Inequality (18) holds for

q\geq M_{n}

$M_{n}$	$3.60\times 10^{12}$	$2.10\times 10^{8}$	$2.60\times 10^{6}$	$222280$	$46892$	$16049$	$7331$
$n$	7	8	9	10	11	12	13
$\nu$	7.8	7.4	7.2	7.1	7.1	7.1	7.1

Now, for each $10\leq n\leq 13,$ and $q<M_{n}$ , by calculating the exact values of $W(x^{n}-1)$ , $W(R)$ and $W(G)$ , and testing Inequality (13) with $\vartheta=2$ , we obtain that, there are $136$ pairs of $(q,n)$ , which fail to satisfy this inequality. For these pairs, we use the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=2$ , which returns False for the pairs $(3,10)$ , $(11,10)$ , $(2,11)$ , $(3,11)$ , $(5,12)$ , $(7,12)$ , and $(13,12)$ . Finally, for these pairs, we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists.

Now, let $n=7$ , then for $q<M_{7}$ , we use Lemma 5.4 with $d=q-1$ , $h=1$ , $H=1$ and $\vartheta=2.$ In this case, the primes $p$ dividing $(q^{7}-1)/d$ , co-prime to $d$ , are of the form $7s+1$ , and $P_{l}\leq\sum_{i=0}^{6}M_{7}^{i}$ for $l\leq 29$ . Further, if $q\geq 65$ , then $\mathcal{D}>0.6608$ and $\mathcal{S}\leq 64.042$ , and there exists a desired pair $(\alpha,\alpha^{-1})$ , if $q^{3/2}>2\mathcal{C}_{\nu}q^{1/\nu}\mathcal{S}$ , which is true for $q\geq 149$ and $\nu=2.8.$ Proceeding in this way for $65\leq q<149$ , we get that, the inequality $q^{3/2}>2\mathcal{C}_{\nu}q^{1/\nu}\mathcal{S}$ holds for all $68\leq q\leq 149.$ Now, for $q<68$ , we use the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=2$ , which returns False for $q=$ 2, 3, 4, 5, 8, 9, 11, 13, and $n=$ 7. In the end, for these pairs, we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists.

Finally, for the cases $n=8$ , 9, and $q<M_{n}$ , again we use Lemma 5.4 with $d=q^{4}-1$ , $q^{3}-1$ , respectively. In these cases, the primes $p$ dividing $(q^{n}-1)/d$ , co-prime to $d$ , are of the form $8s+1$ , when $n=8$ , and of the form $9s+1$ , when $n=9$ . Now, following the same line as in the case $n=7$ , we have reduced the bounds on $q$ from $M_{8}$ to 203, and from $M_{9}$ to 41.Now, for $q<68$ , we use the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=2$ , which returns False for $q=$ 3, 5, 7, 9, 11, 13, 17 and $n=$ 8, and $q=$ 2, 4, 7, 19, and $n=9$ . Finally, for these pairs, we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists. ∎

Lemma 5.6.

Let $7\leq n\leq 13$ and $q$ be a prime power such that $\mathrm{gcd}(q,n)\neq 1$ . Then, there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ .

Proof.

Since, $\mathrm{gcd}(q,n)\neq 1$ , $q=p^{t}$ , for each prime $p$ dividing $n$ and some positive integer $t$ , and Inequality (13) holds, if

p^{nt/2-3}>\mathcal{C}_{\nu}p^{nt/\nu}2^{2n}

(19)

For each $7\leq n\leq 13$ , the above inequality holds for $t\geq t_{n,p}$ and the suitable choices of $\nu$ , listed in Table 5.

Table 5: Values of

t_{n,p}

such that Inequality (19) holds for

t\geq t_{n,p}

$t_{n,p}$	$3797$	$223$	$46$	$42,19$	$9$	$24,15$	$6$
$(n,p)$	$(7,7)$	$(8,2)$	$(9,3)$	$(10,2),(10,5)$	$(11,11)$	$(12,2),(12,3)$	$(13,13)$
$\nu$	15.7	10.3	9	8	7.5	7.5	7

For the cases $9\leq n\leq 13$ , and $t<t_{n,p}$ , by calculating the exact values of $W(x^{n}-1)$ , $W(R)$ and $W(G)$ , and testing Inequality (13) with $\vartheta=3$ , we obtain that, there are $25$ pairs of $(q,n)$ , which fail to satisfy this inequality. For these pairs, we use the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=3$ , which returns False for the pairs $(3,9)$ , $(9,9)$ , $(2,10)$ , $(4,10)$ , $(16,10)$ , $(5,10)$ , $(2,12)$ , $(4,12)$ , $(3,12)$ and $(9,12)$ . Finally, for these pairs, we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists.

Now, let $n=7$ , $q=7^{t}$ and $t<t_{7,7}$ . Clearly, $x^{7}-1$ has only one linear factor over $\mathbb{F}_{q}$ and $7\nmid q^{7}-1$ , therefore, using Lemma 4.4 with $d=q-1$ , $h=1$ and $H=1$ , we get that there exists a pair $(\alpha,\alpha^{-1})$ of desired properties, if $7^{t/2}>2\mathcal{C}_{\nu}7^{t/\nu}\mathcal{S}$ . Since, the primes dividing $(q^{7}-1)/d$ are of the form $7s+1$ , so let $P_{l}$ and $S_{l}$ be the product and sum of reciprocals of first $l$ primes of the form $7s+1$ respectively, such that $P_{l}\leq\sum_{i=0}^{6}7^{it}$ , then $l\leq 3891$ , and from Lemma 4.4, $\mathcal{D}\geq 1-S_{l}-1/7^{t}$ and $\mathcal{S}\leq l/D+2.$ Now, For $t\geq 8$ , we get $\mathcal{D}>0.7661$ and $\mathcal{S}\leq 5080.82$ , and inequality $7^{t/2}>2\mathcal{C}_{\nu}7^{t/\nu}\mathcal{S}$ holds for $t\geq 20$ and $\nu=5$ . Proceeding in this way for $8\leq t<20$ , we get that, the inequality $7^{t/2}>2\mathcal{C}_{\nu}7^{t/\nu}\mathcal{S}$ always holds for $t\geq 11.$ In the end, for $1\leq t\leq 10$ , we first perform the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=3$ , which returns False only for $t=$ 1, 2, 3, 4, and then for these values, we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists.

Now, in the case $n=8$ , we have $q=2^{t}$ and $x^{8}-1=(x-1)^{8}$ over $\mathbb{F}_{q}.$ If $d=q^{4}-1$ and $p$ is a prime divisor of $q^{4}+1$ co-prime to $d$ , then $q^{4}\not\equiv 1(\mathrm{mod}\hskip 1.42262ptp)$ and $q^{4}\equiv-1(\mathrm{mod}\hskip 1.42262ptp)$ , which implies that, the multiplicative order of $q$ modulo $p$ is $8$ . Thus, $p$ is of the form $8s+1.$ Again, we use Lemma 4.4 with $d=q^{4}-1$ , $h=H=1$ . Let $P_{l}$ and $S_{l}$ be the product and sum of reciprocals of first $l$ primes of the form $8s+1$ respectively, such that $P_{l}\leq 2^{4t}+1$ , then $l\leq 90$ for $t<t_{8,2}$ , and from Lemma 4.4, $\mathcal{D}\geq 1-S_{l}-1/2^{t}$ and $\mathcal{S}\leq l/D+2.$ Now, For $t\geq 8$ , we get $\mathcal{D}>0.764$ and $\mathcal{S}\leq 119.75$ , and inequality $2^{t}>2\mathcal{C}_{\nu}2^{4t/\nu}\mathcal{S}$ holds for $t\geq 35$ and $\nu=7$ . Proceeding in this way for $8\leq t<35$ , we get that, the inequality $2^{t}>2\mathcal{C}_{\nu}q^{4t/\nu}\mathcal{S}$ always holds for $t\geq 30.$ In the end, for $1\leq t\leq 29$ , we obtain that, there are 14 values of $t$ ( $1\leq t\leq 12;\ t=15$ , 18) which fails to satisfy Inequality (13). For these values, we use the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=3$ , which returns False for $1\leq t\leq 6$ . Finally, we explicitly search for a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ using SageMath procedure DIRECT_SEARCH(q,n), and find that such a pair always exists. ∎

Lemma 5.7.

Let $n=$ $5$ , $6$ , and $q$ be a prime power such that $\mathrm{gcd}(q,n)=1$ . Then, there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ with the sole genuine exception $(4,5)$ .

Proof.

First, we test Inequality (18) in both the cases $n=$ 5, 6, and get that, this holds for $q\geq M_{n}$ , where $M_{5}=6.97\times 10^{323}$ , if we set $\nu=11.9$ , and $M_{6}=4.74\times 10^{27}$ , if we set $\nu=8.7.$ Now, for $q<M_{n}$ , in each case, we use Lemma 5.4 with $d=q-1$ for $n=5$ , and $d=q^{2}-1$ for $n=6.$ Notice that, primes dividing $(q^{n}-1)/d$ , co-prime to $d$ , are of the form $5s+1$ for $n=5$ , and $3s+1$ for $n=6.$ Then, for $n=5$ and $q\geq 10^{5}$ , we get $l\leq 360$ , $\mathcal{D}>0.648$ , $\mathcal{S}\leq 569.9$ , and for $n=6$ and $q\geq 10^{4}$ , we get $l\leq 48$ , $\mathcal{D}>0.429$ , $\mathcal{S}\leq 137.11.$ With these, we test the inequalities $q^{1/2}>2\mathcal{C}_{\nu}q^{1/\nu}\mathcal{S}$ for $n=5;\ \nu=5.5$ , and $q>2\mathcal{C}_{\nu}q^{2/\nu}\mathcal{S}$ for $n=6;\ \nu=5.2$ , and get that, there always exists a pair $(\alpha,\alpha^{-1})$ of desired properties for $q\geq 4.395\times 10^{13}$ and $n=5$ , and $q\geq 646882$ and $n=6.$ By repeating the process for remaining values of $q$ in each case, we significantly reduce the bound on $q$ from $M_{5}$ to $4.96\times 10^{9}$ , and from $M_{6}$ to $94531.$

Now, for $n=5;$ $6$ and $q<4.96\times 10^{9};$ $94531$ , respectively, we use Lemma 5.4 with $d=\mathrm{gcd}(2\cdot 3\cdot 5,q^{n}-1)$ . Notice that, the primes dividing $(q^{n}-1)/d$ co-prime to $d$ are of the form $2s+1$ in each case. Then, for $10^{5}\leq q<4.96\times 10^{9}$ and $n=5$ , we get $l\leq 28$ , $\mathcal{D}>0.175$ , $\mathcal{S}\leq 207.05$ , and the inequality $q^{1/2}>2^{4}\mathcal{S}$ holds for $q\geq 10973928$ . Similarly, for $10^{3}\leq q<94531$ and $n=6$ , we get $l\leq 19$ , $\mathcal{D}>0.253$ , $\mathcal{S}\leq 116.58$ , and the inequality $q>2^{4}\mathcal{S}$ holds for $q\geq 1866$ . Proceeding in this way, we get that, there always exists a pair $(\alpha,\alpha^{-1})$ of desired properties for $q\geq 3839356$ and $n=5$ , and $q\geq 1109$ and $n=6$ . For the remaining values of $q$ in each case, we run the SageMath procedure TEST_SIEVE(q,n, $\vartheta$ ) with $\vartheta=2$ and get that, there are $871$ values of $q$ (listed in Table LABEL:Table6) in the case of $n=5$ , and $47$ values of $q$ (listed in Table LABEL:Table6) in the case of $n=6$ , for which this procedure returns False. Finally, we perform the procedure DIRECT_SEARCH(q,n) for the remaining values of $t$ , and find that, there always exists the desired pairs $(\alpha,\alpha^{-1})$ with the sole genuine exception $(4,5).$ ∎

Summing up Lemmas 5.1, 5.3, 5.5, 5.6, and 5.7, we conclude our Theorem 1.3.

5.3 A note on the cases $n=1,2,3,4$

From the definition of $k$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , it is clear that $0\leq k\leq n-1$ , therefore, it is sensible to talk about the existence of a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ for $n>1$ . Now, for the case $n=2$ , if $\alpha$ is a $1$ -normal element in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ , then either $\alpha^{q}-\alpha=0$ or $\alpha^{q}+\alpha=0$ , which implies $\mathrm{ord}(\alpha)\leq 2(q-1)<q^{2}-1$ . Thus, there does not exist any pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{2}}$ over $\mathbb{F}_{q}.$ Further, we show that, such a pair does not exists in the case of $n=3$ also.

Lemma 5.8.

Let $n=3$ and $q\geq 2$ be any prime power. Then, there does not exist any pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}.$

Proof.

Suppose that, there exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{3}}$ over $\mathbb{F}_{q}$ . Since, $x^{3}-1=(x-1)^{3}$ or $(x-1)(x^{2}+x+1)$ or $(x-1)(x-a)(x-a^{-1})$ for some $a\in\mathbb{F}_{q}\setminus\{0,1\}$ . Therefore, we have the following cases on the $\mathbb{F}_{q}$ -orders of $\alpha$ and $\alpha^{-1}$ .

Case 1. If $\mathrm{Ord}_{q}(\alpha)=\mathrm{Ord}_{q}(\alpha^{-1})=x^{2}+x+1$ , then, the trace of $\alpha$ and $\alpha^{-1}$ over $\mathbb{F}_{q}$ is equal to zero. Consequently, the primitive polynomial of $\alpha$ will be of the form $x^{3}+a$ for some $a\in\mathbb{F}_{q},$ and hence $\alpha^{3}\in\mathbb{F}_{q}.$ This means $q^{3}-1\mid 3(q-1)$ , which is never possible for any $q>1$ .

Case 2. If $(x-1)\mid\mathrm{gcd}(\mathrm{Ord}_{q}(\alpha),\mathrm{Ord}_{q}(\alpha^{-1}))$ , then we get that, $(\alpha^{q}-\alpha)^{q-1}$ and $(\alpha^{-q}-\alpha^{-1})^{q-1}$ both belong to $\mathbb{F}_{q}.$ This implies $\alpha^{q^{2}-1}\in\mathbb{F}_{q}$ , i.e. $q^{3}-1\mid(q^{2}-1)(q-1)$ , which is not possible.

Case 3. If $\mathrm{Ord}_{q}(\alpha)=(x-1)(x-a)$ and $\mathrm{Ord}_{q}(\alpha^{-1})=(x-a)(x-a^{-1})$ , then we get that, $(\alpha^{q}-a\alpha)^{q-1}=1$ and $(\alpha^{-q}-a^{-1}\alpha^{-1})^{q-1}\in\mathbb{F}_{q}.$ This implies $\alpha^{q^{2}-1}\in\mathbb{F}_{q}$ , i.e. $q^{3}-1\mid(q^{2}-1)(q-1)$ , which is not possible.

Case 4. If $\mathrm{Ord}_{q}(\alpha)=(x-1)(x-a^{-1})$ and $\mathrm{Ord}_{q}(\alpha^{-1})=(x-a)(x-a)$ ,then by interchanging the role of $\alpha$ and $\alpha^{-1}$ in Case 3, we get that, $q^{3}-1\mid(q^{2}-1)(q-1)$ , which is not possible.

This completes the proof of the lemma. ∎

We also provide a necessary condition for the existence of a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{4}}$ over $\mathbb{F}_{q}.$

Lemma 5.9.

If there exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{4}}$ over $\mathbb{F}_{q}$ , then $q\equiv 1\hskip 1.42262pt(\mathrm{mod}\hskip 1.42262pt).$

Proof.

Let’s assume that $q\not\equiv 1\hskip 1.42262pt(\mathrm{mod}\hskip 1.42262pt)$ , then the polynomial $x^{4}-1$ factorizes either as $(x-1)^{4}$ , or as $(x-1)(x+1)(x^{2}+1)$ over $\mathbb{F}_{q}.$ Further, let $(\alpha,\alpha^{-1})$ be a pair of $1$ -normal elements in $\mathbb{F}_{q^{4}}$ over $\mathbb{F}_{q}$ . Then, we have the following cases.

Case 1: $(x-1)\mid\mathrm{gcd}(\mathrm{Ord}_{q}(\alpha),\mathrm{Ord}_{q}(\alpha^{-1})),$ then $\alpha^{q^{2}}-\alpha$ and $\alpha^{-q^{2}}-\alpha^{-1}$ both will belong to $\mathbb{F}_{q}$ . This implies $\alpha^{q^{2}+1}\in\mathbb{F}_{q}$ , i.e. $\mathrm{ord}(\alpha)\leq(q^{2}+1)(q-1)<q^{4}-1.$

Case 2: $(x+1)\mid\mathrm{gcd}(\mathrm{Ord}_{q}(\alpha),\mathrm{Ord}_{q}(\alpha^{-1})),$ then $(\alpha^{q^{2}}-\alpha)^{2}$ and $(\alpha^{-q^{2}}-\alpha^{-1})^{2}$ both will belong to $\mathbb{F}_{q}$ . This implies $\alpha^{2(q^{2}+1)}\in\mathbb{F}_{q}$ , i.e. $\mathrm{ord}(\alpha)\leq 2(q^{2}+1)(q-1)<q^{4}-1.$

Case 3: $\mathrm{gcd}(\mathrm{Ord}_{q}(\alpha),\mathrm{Ord}_{q}(\alpha^{-1}))=x^{2}+1$ , then either $\alpha^{q^{2}}-\alpha$ and $(\alpha^{-q^{2}}-\alpha^{-1})^{2}$ belong to $\mathbb{F}_{q}$ , or $(\alpha^{q^{2}}-\alpha)^{2}$ and $\alpha^{-q^{2}}-\alpha^{-1}$ will belong to $\mathbb{F}_{q}$ simultaneously. In either case, we get that $\mathrm{ord}(\alpha)\leq 2(q^{2}+1)(q-1)<q^{4}-1.$

In each case, we get that $\mathrm{ord}(\alpha)<q^{4}-1$ , which implies $\alpha$ is never a primitive element in $\mathbb{F}_{q}.$ Thus, a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal elements in $\mathbb{F}_{q^{4}}$ over $\mathbb{F}_{q}$ can exist, only if $q\equiv 1\hskip 1.42262pt(\mathrm{mod}\hskip 1.42262pt4).$ ∎

For the cases, $n=4$ and $q=4t+1$ ; $n=5$ and $q=5^{t}$ ; and $n=6$ and $q=2^{t},3^{t}$ , where $t$ is a positive integer, we performed the extensive experiments, and obtained that, there always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ unless $n=6$ and $q=2,4$ . On the basis of these experiments, we conjecture the following.

Conjecture 5.1.

Let $t$ be a positive integer. There always exists a pair $(\alpha,\alpha^{-1})$ of primitive $1$ -normal in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ if $n=4$ and $q=4t+1;$ $n=5$ and $q=5^{t};$ and $n=6$ and $q=2^{t},3^{t}$ with the sole genuine exceptions $(2,6)$ , $(4,6)$ .

6 Acknowledgements

This work was funded by Council of Scientific $\&$ Industrial Research, under Grant F. No. 09/045(1674)/2019-EMR-I and University Grant Commission, under Grant Ref. No. 1042/CSIR-UGC NET DEC-2018.

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Appendix A

Table 6: Pairs

(q,n)

for which TEST_SIEVE(q,n) returns False in the cases

n=

5, 6.

$n$	$q$
5	2, 3, 4, 7, 8, 9, 11, 13, 16, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281, 283, 289, 293, 307, 311, 313, 317, 331, 337, 343, 347, 349, 353, 359, 361, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 512, 521, 523, 529, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 729, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 841, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 961, 967, 971, 977, 991, 997, 1009, 1013, 1019, 1021, 1024, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1193, 1201, 1213, 1229, 1231, 1237, 1249, 1259, 1279, 1289, 1291, 1301, 1303, 1321, 1327, 1331, 1361, 1367, 1369, 1373, 1381, 1399, 1409, 1423, 1429, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1489, 1499, 1511, 1531, 1543, 1549, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1609, 1619, 1621, 1627, 1667, 1669, 1681, 1693, 1697, 1699, 1709, 1721, 1723, 1741, 1747, 1753, 1759, 1777, 1783, 1789, 1801, 1811, 1831, 1849, 1861, 1867, 1871, 1873, 1879, 1901, 1931, 1933, 1949, 1951, 1993, 1999, 2003, 2011, 2017, 2029, 2053, 2081, 2083, 2087, 2089, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2161, 2179, 2197, 2203, 2209, 2221, 2239, 2251, 2269, 2281, 2293, 2297, 2311, 2341, 2347, 2351, 2371, 2377, 2381, 2389, 2401, 2411, 2437, 2441, 2467, 2473, 2503, 2521, 2531, 2539, 2551, 2557, 2591, 2593, 2621, 2647, 2659, 2671, 2683, 2689, 2699, 2707, 2711, 2713, 2719, 2731, 2741, 2749, 2767, 2791, 2797, 2801, 2803, 2809, 2851, 2857, 2861, 2887, 2927, 2953, 2971, 3001, 3011, 3019, 3037, 3041, 3049, 3061, 3067, 3079, 3109, 3121, 3169, 3181, 3191, 3217, 3221, 3251, 3259, 3271, 3301, 3319, 3331, 3361, 3371, 3391, 3433, 3457, 3461, 3463, 3469, 3481, 3491, 3499, 3511, 3529, 3541, 3547, 3571, 3581, 3613, 3631, 3671, 3691, 3697, 3701, 3721, 3727, 3733, 3739, 3761, 3821, 3823, 3851, 3853, 3877, 3881, 3907, 3911, 3919, 3931, 4001, 4003, 4019, 4021, 4051, 4057, 4096, 4111, 4129, 4159, 4201, 4211, 4219, 4229, 4231, 4241, 4243, 4261, 4271, 4327, 4391, 4421, 4441, 4447, 4451, 4481, 4489, 4519, 4561, 4591, 4603, 4621, 4651, 4691, 4751, 4759, 4789, 4801, 4831, 4861, 4871, 4909, 4931, 4951, 4957, 4999, 5011, 5021, 5041, 5101, 5119, 5167, 5171, 5179, 5209, 5261, 5281, 5329, 5351, 5381, 5419, 5431, 5441, 5449, 5471, 5501, 5503, 5521, 5531, 5581, 5591, 5641, 5659, 5701, 5741, 5743, 5779, 5791, 5801, 5821, 5839, 5851, 5861, 5881, 5923, 5981, 6007, 6011, 6043, 6091, 6121, 6131, 6151, 6163, 6211, 6221, 6241, 6271, 6301, 6361, 6421, 6427, 6451, 6469, 6481, 6521, 6561, 6571, 6581, 6637, 6661, 6679, 6691, 6761, 6763, 6781, 6791, 6841, 6871, 6889, 6961, 6991, 7001, 7039, 7121, 7151, 7177, 7309, 7321, 7351, 7411, 7417, 7451, 7481, 7541, 7549, 7561, 7591, 7621, 7681, 7741, 7841, 7921, 7951, 8011, 8101, 8161, 8171, 8191, 8221, 8231, 8311, 8317, 8419, 8431, 8461, 8501, 8521, 8581, 8641, 8731, 8737, 8761, 8779, 8821, 8941, 8971, 9001, 9091, 9151, 9161, 9181, 9199, 9241, 9311, 9391, 9409, 9421, 9491, 9511, 9521, 9601, 9619, 9631, 9661, 9721, 9769, 9781, 9811, 9829, 9871, 9901, 9931, 10111, 10141, 10151, 10201, 10321, 10459, 10501, 10531, 10609, 10651, 10711, 10771, 10831, 10861, 10891, 11047, 11071, 11131, 11161, 11251, 11257, 11311, 11411, 11491, 11551, 11621, 11701, 11719, 11731, 11821, 11881, 11941, 11971, 12211, 12241, 12301, 12391, 12421, 12433, 12451, 12511, 12541, 12601, 12721, 12769, 12781, 12841, 13171, 13291, 13381, 13399, 13411, 13441, 13567, 13681, 13711, 13831, 13921, 14071, 14221, 14251, 14281, 14341, 14401, 14431, 14551, 14641, 14731, 14771, 14821, 14851, 15031, 15121, 15271, 15331, 15361, 15391, 15451, 15511, 15541, 15601, 15661, 15901, 15991, 16111, 16141, 16231, 16339, 16381, 16651, 16831, 16921, 17011, 17161, 17191, 17341, 17431, 17491, 17581, 17791, 17851, 17911, 18061, 18121, 18181, 18301, 18451, 18481, 18661, 18691, 19081, 19141, 19231, 19321, 19381, 19441, 19471, 19501, 19531, 19801, 19891, 20011, 20101, 20161, 20431, 20551, 20641, 20749, 21001, 21121, 21211, 21481, 21751, 21841, 21871, 21961, 22051, 22111, 22201, 22291, 22441, 22531, 22621, 22741, 22801, 22861, 23011, 23071, 23131, 23311, 23371, 23431, 23761, 23971, 24061, 24091, 24121, 24151, 24181, 24391, 24421, 24481, 24571, 24781, 24841, 25111, 25261, 25411, 25621, 25741, 26041, 26251, 26701, 26821, 26881, 27031, 27091, 27361, 27691, 27751, 27901, 28051, 28081, 28351, 28561, 28711, 28771, 29131, 29191, 29401, 29581, 29611, 29641, 29671, 29881, 30211, 30241, 30391, 30661, 30871, 31081, 31321, 31531, 31981, 32191, 32341, 32371, 32761, 32971, 33091, 33181, 33391, 33811, 34171, 34651, 35281, 35311, 35491, 36061, 36481, 36541, 37171, 37591, 37951, 38611, 38851, 39901, 40111, 40531, 41611, 42331, 42841, 43261, 43891, 44521, 46411, 47881, 51871, 53551, 57121, 60901, 62791, 63211, 65521, 70981
6	5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 169, 179, 181, 191, 199, 211, 229, 241.

n

q

2, 3, 4, 7, 8, 9, 11, 13, 16, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281, 283, 289, 293, 307, 311, 313, 317, 331, 337, 343, 347, 349, 353, 359, 361, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 512, 521, 523, 529, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 729, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 841, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 961, 967, 971, 977, 991, 997, 1009, 1013, 1019, 1021, 1024, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1193, 1201, 1213, 1229, 1231, 1237, 1249, 1259, 1279, 1289, 1291, 1301, 1303, 1321, 1327, 1331, 1361, 1367, 1369, 1373, 1381, 1399, 1409, 1423, 1429, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1489, 1499, 1511, 1531, 1543, 1549, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1609, 1619, 1621, 1627, 1667, 1669, 1681, 1693, 1697, 1699, 1709, 1721, 1723, 1741, 1747, 1753, 1759, 1777, 1783, 1789, 1801, 1811, 1831, 1849, 1861, 1867, 1871, 1873, 1879, 1901, 1931, 1933, 1949, 1951, 1993, 1999, 2003, 2011, 2017, 2029, 2053, 2081, 2083, 2087, 2089, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2161, 2179, 2197, 2203, 2209, 2221, 2239, 2251, 2269, 2281, 2293, 2297, 2311, 2341, 2347, 2351, 2371, 2377, 2381, 2389, 2401, 2411, 2437, 2441, 2467, 2473, 2503, 2521, 2531, 2539, 2551, 2557, 2591, 2593, 2621, 2647, 2659, 2671, 2683, 2689, 2699, 2707, 2711, 2713, 2719, 2731, 2741, 2749, 2767, 2791, 2797, 2801, 2803, 2809, 2851, 2857, 2861, 2887, 2927, 2953, 2971, 3001, 3011, 3019, 3037, 3041, 3049, 3061, 3067, 3079, 3109, 3121, 3169, 3181, 3191, 3217, 3221, 3251, 3259, 3271, 3301, 3319, 3331, 3361, 3371, 3391, 3433, 3457, 3461, 3463, 3469, 3481, 3491, 3499, 3511, 3529, 3541, 3547, 3571, 3581, 3613, 3631, 3671, 3691, 3697, 3701, 3721, 3727, 3733, 3739, 3761, 3821, 3823, 3851, 3853, 3877, 3881, 3907, 3911, 3919, 3931, 4001, 4003, 4019, 4021, 4051, 4057, 4096, 4111, 4129, 4159, 4201, 4211, 4219, 4229, 4231, 4241, 4243, 4261, 4271, 4327, 4391, 4421, 4441, 4447, 4451, 4481, 4489, 4519, 4561, 4591, 4603, 4621, 4651, 4691, 4751, 4759, 4789, 4801, 4831, 4861, 4871, 4909, 4931, 4951, 4957, 4999, 5011, 5021, 5041, 5101, 5119, 5167, 5171, 5179, 5209, 5261, 5281, 5329, 5351, 5381, 5419, 5431, 5441, 5449, 5471, 5501, 5503, 5521, 5531, 5581, 5591, 5641, 5659, 5701, 5741, 5743, 5779, 5791, 5801, 5821, 5839, 5851, 5861, 5881, 5923, 5981, 6007, 6011, 6043, 6091, 6121, 6131, 6151, 6163, 6211, 6221, 6241, 6271, 6301, 6361, 6421, 6427, 6451, 6469, 6481, 6521, 6561, 6571, 6581, 6637, 6661, 6679, 6691, 6761, 6763, 6781, 6791, 6841, 6871, 6889, 6961, 6991, 7001, 7039, 7121, 7151, 7177, 7309, 7321, 7351, 7411, 7417, 7451, 7481, 7541, 7549, 7561, 7591, 7621, 7681, 7741, 7841, 7921, 7951, 8011, 8101, 8161, 8171, 8191, 8221, 8231, 8311, 8317, 8419, 8431, 8461, 8501, 8521, 8581, 8641, 8731, 8737, 8761, 8779, 8821, 8941, 8971, 9001, 9091, 9151, 9161, 9181, 9199, 9241, 9311, 9391, 9409, 9421, 9491, 9511, 9521, 9601, 9619, 9631, 9661, 9721, 9769, 9781, 9811, 9829, 9871, 9901, 9931, 10111, 10141, 10151, 10201, 10321, 10459, 10501, 10531, 10609, 10651, 10711, 10771, 10831, 10861, 10891, 11047, 11071, 11131, 11161, 11251, 11257, 11311, 11411, 11491, 11551, 11621, 11701, 11719, 11731, 11821, 11881, 11941, 11971, 12211, 12241, 12301, 12391, 12421, 12433, 12451, 12511, 12541, 12601, 12721, 12769, 12781, 12841, 13171, 13291, 13381, 13399, 13411, 13441, 13567, 13681, 13711, 13831, 13921, 14071, 14221, 14251, 14281, 14341, 14401, 14431, 14551, 14641, 14731, 14771, 14821, 14851, 15031, 15121, 15271, 15331, 15361, 15391, 15451, 15511, 15541, 15601, 15661, 15901, 15991, 16111, 16141, 16231, 16339, 16381, 16651, 16831, 16921, 17011, 17161, 17191, 17341, 17431, 17491, 17581, 17791, 17851, 17911, 18061, 18121, 18181, 18301, 18451, 18481, 18661, 18691, 19081, 19141, 19231, 19321, 19381, 19441, 19471, 19501, 19531, 19801, 19891, 20011, 20101, 20161, 20431, 20551, 20641, 20749, 21001, 21121, 21211, 21481, 21751, 21841, 21871, 21961, 22051, 22111, 22201, 22291, 22441, 22531, 22621, 22741, 22801, 22861, 23011, 23071, 23131, 23311, 23371, 23431, 23761, 23971, 24061, 24091, 24121, 24151, 24181, 24391, 24421, 24481, 24571, 24781, 24841, 25111, 25261, 25411, 25621, 25741, 26041, 26251, 26701, 26821, 26881, 27031, 27091, 27361, 27691, 27751, 27901, 28051, 28081, 28351, 28561, 28711, 28771, 29131, 29191, 29401, 29581, 29611, 29641, 29671, 29881, 30211, 30241, 30391, 30661, 30871, 31081, 31321, 31531, 31981, 32191, 32341, 32371, 32761, 32971, 33091, 33181, 33391, 33811, 34171, 34651, 35281, 35311, 35491, 36061, 36481, 36541, 37171, 37591, 37951, 38611, 38851, 39901, 40111, 40531, 41611, 42331, 42841, 43261, 43891, 44521, 46411, 47881, 51871, 53551, 57121, 60901, 62791, 63211, 65521, 70981

5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 169, 179, 181, 191, 199, 211, 229, 241.

Algorithm 1 Testing the sieving inequality for the existence of primitive

1

-normal pair

(\alpha,\alpha^{-1})

q

n

\vartheta

2:True, if

q^{n/2-\vartheta}>2W(h)W(d)W(H)\mathcal{S}

for some

h\mid x^{n}-1

d\mid R

and

H\mid G

otherwise, False.

3:procedure test_sieve(q,n,

\vartheta

)

4: Construct the set

A

of divisors of

\mathrm{rad}(q^{n}-1)

and set

B

of divisors of

\frac{\mathrm{rad}(x^{n}-1)}{x-1}

5: for

d

A

6: for

H

B

h=H

8: Construct the set

L_{1}

of prime divisors

\frac{\mathrm{rad}(q^{n}-1)}{d}

9: Construct the set

L_{2}

of irreducible factors of

\frac{\mathrm{rad}(x^{n}-1)}{H(x-1)}

and the set

L_{3}=L_{2}\cup\{x-1\}

10: Compute

\mathcal{D}=1-\sum\limits_{p_{1}\in L_{1}}\frac{1}{p_{1}}-\sum\limits_{p_{2}\in L_{2}}\frac{1}{q^{\mathrm{deg}(p_{2})}}-\sum\limits_{p_{3}\in L_{3}}\frac{1}{q^{\mathrm{deg}(p_{3})}}

11: if

\mathcal{D}>0

then

12: compute

\mathcal{S}=\frac{\ |L_{1}|+|L_{2}|+|L_{3}|-1}{\mathcal{D}}+2

13: if

q^{n/2-\vartheta}>2W(d)W(H)^{2}\mathcal{S}

then

\triangleright

Sieving inequality

14: return True

15: end if

16: end if

17: end for

18: end for

19: return False

20:end procedure

Algorithm 2 Explicit search for a primitive

1

-normal pair

(\alpha,\alpha^{-1})

q

n

2:True, if a pair

(\alpha,\alpha^{-1})

of primitive

1

-normal elements exists in

\mathbb{F}_{q^{n}}

over

\mathbb{F}_{q}

, otherwise, False.

3:procedure direct_search(q,n)

4: for

\beta

\mathbb{F}_{q^{n}}

\alpha=\beta^{q}-\beta

6: if

\alpha\neq 0

then

7: construct

m_{\alpha}(x)

and

m_{\alpha^{-1}}(x)

\triangleright

m_{\alpha}(x)=\sum_{i=0}^{n-1}\alpha^{q^{i}}x^{n-i-1}

8: if

\mathrm{deg}(\mathrm{gcd}(m_{\alpha}(x),x^{n}-1))=1

\mathrm{deg}(\mathrm{gcd}(m_{\alpha^{-1}}(x),x^{n}-1))=1

\mathrm{ord}(\alpha)=q^{n}-1

then

9: return True

10: end if

11: end if

12: end for

13: return False

14:end procedure

	$\displaystyle\big{\|}N_{r,k,g}(h,d,H)-$	$\displaystyle(q^{n}-q^{k})\bm{\ell}\big{(}(1)_{j=1}^{s},(1)_{i=1}^{t}\big{)}\mathcal{H}\big{\|}\leq$
		$\displaystyle\mathcal{H}\underbrace{\sum_{\begin{subarray}{c}h^{\prime}\mid h\end{subarray}}\sum_{\begin{subarray}{c}d_{1}\mid d,d_{2}\mid u,\\ e_{j}\mid\lambda_{j};1\leq j\leq s\end{subarray}}\sum_{\begin{subarray}{c}g_{1}\mid H,g_{2}\mid\pi,\\ h_{i},\mid\Lambda_{i};1\leq i\leq t\end{subarray}}}_{\text{(all are not 1 simultaneously)}}\big{\|}\frac{\bm{\mu}}{\bm{\Phi}}(h^{\prime},d_{1},g_{1})\big{\|}\big{\|}\bm{\ell}\big{(}(e_{j})_{j=1}^{s},(h_{i})_{i=1}^{t}\big{)}\big{\|}\times$
		$\displaystyle\sum_{\begin{subarray}{c}\psi_{h}^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}\chi_{d_{1}},\chi_{d_{2}},\\ \chi_{e_{j}};1\leq j\leq s\end{subarray}}\sum_{\begin{subarray}{c}\psi_{g_{1}},\psi_{g_{2}},\\ \psi_{h_{i}};\ 1\leq i\leq t\end{subarray}}2q^{n/2+k}.$

Inverses of rr-primitive kk-normal elements over finite fields

Abstract

keywords:

MSC:

1 Introduction

Definition 1.1.

Theorem 1.1.

Theorem 1.2 (Strong Primitive Normal Basis Theorem).

Theorem 1.3.

2 Preliminaries

2.1 Freeness

Definition 2.1 (ee-free element).

Definition 2.2.

Definition 2.3.

Definition 2.4.

Lemma 2.1.

2.2 Characters

Lemma 2.2.

Lemma 2.3.

Lemma 2.4.

2.3 Some characteristic functions

3 Characterization of rr-primitive and kk-normal elements

3.1 Characteristic function for the set of rr-primitive elements

3.2 Characteristic function for the set of kk-normal elements

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

4 Existence of an rr-primitive kk-normal pair (α,α−1)(\alpha,\alpha^{-1})

Theorem 4.1.

Proof.

Corollary 4.1.1.

Proof.

Lemma 4.1.

Lemma 4.2.

Proposition 4.1.

Proof.

4.1 The sieving technique

Lemma 4.3.

Lemma 4.4.

5 Existence of a primitive 11-normal pair (α,α−1)(\alpha,\alpha^{-1})

5.1 The cases n≥14n\geq 14

Lemma 5.1.

Proof.

Lemma 5.2.

Lemma 5.3.

Proof.

5.2 The cases 5≤n≤135\leq n\leq 13

Lemma 5.4.

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

Lemma 5.7.

Proof.

5.3 A note on the cases n=1,2,3,4n=1,2,3,4

Lemma 5.8.

Proof.

Lemma 5.9.

Proof.

Conjecture 5.1.

6 Acknowledgements

References

Appendix A

Inverses of $r$ -primitive $k$ -normal elements over finite fields

Definition 2.1 ( $e$ -free element).

3 Characterization of $r$ -primitive and $k$ -normal elements

3.1 Characteristic function for the set of $r$ -primitive elements

3.2 Characteristic function for the set of $k$ -normal elements

4 Existence of an $r$ -primitive $k$ -normal pair $(\alpha,\alpha^{-1})$

5 Existence of a primitive $1$ -normal pair $(\alpha,\alpha^{-1})$

5.1 The cases $n\geq 14$

5.2 The cases $5\leq n\leq 13$

5.3 A note on the cases $n=1,2,3,4$