Construction of Permutation Polynomials Using Additive and Multiplicative Characters

Abstract

Permutation is a natural phenomenon useful for understanding and explaining the structural and functional behavior of objects or concepts. The mathematical formulation of permutation behavior can be readily achieved by permutation polynomials. Permutation polynomials are constructed by suitably modifying linearized polynomials and associated affine polynomials with the help of additive characters, multiplicative characters, and special types of Trace functions for the polynomials in one and more than one indeterminates. The permutation properties of the obtained polynomials are verified using the AGW criterion.

1. Introduction

Some usual and natural processes such as ordering and re-ordering, arrangements, and derangements make a remarkable difference in many real-life situations. Various scientific developments require a sensible study of combinatorial designs involving permutations and combinations, which lead to a special branch of mathematics. Over the last 50 years, the theory of finite fields has been a widely discussed branch of algebra due to its diverse application in coding theory, cryptography, and various number-theoretic and combinatorial aspects. Starting from the 17th century, the subject has taken its own part in mathematical concepts, including theoretical, computational, scientific, and technical interests. Eminent mathematicians like Fermat, Euler, Lagrange, and Legendre contributed to the structure theory of finite prime fields. Carl Friedrich Gauss and Evariste Galois worked independently in making the classical general theory of finite fields, which exposed remarkably to the emergence of discrete mathematics and modern number theory as serious algebraic serious disciplines.

Investigation and various applications of finite fields require the theoretic study of polynomials over finite fields as an important tool. It is possible to express any non-constant polynomial over a finite field as a product of all irreducible polynomials. It is required to devise some algorithms which are reasonably efficient for the actual computation of irreducible factors of any polynomial of the positive degree over a finite fields. Beyond the concern of finite fields, the factorization of polynomials is also useful in many algebraic and number-theoretic computational problems in one or the other way.

It is possible to represent any function mapped from a finite field to itself by a polynomial over a finite field. Thus, finite fields are polynomially complete. It is obviously natural to expect the permutation of a finite set of elements in combinatorial study or any such studies in discrete mathematics. The advantage of finite fields over other finite sets is due to the scope of the algebraic study of permutation of elements, which can be achieved through polynomial permutations. Hermite started the initial study of permutation polynomials over finite prime fields and the concept has been continuously explored by many researchers in recent years, including the construction of permutation polynomials and application of permutation polynomials to finite projective geometry or various such mathematical concepts as well as cryptographic applications of permutation polynomials. The most important step was taken by Zieves [1] and his lemma became a powerful tool to construct many old and new kinds of permutation polynomials.

2. Preliminary

A polynomial $f(x)\in F_q[x]$ is called a permutation polynomial over the finite field $F_q$ if the associated polynomial with it is a mapping $f:c\to f(c)$ from $F_q$ to $F_q$ is a bijection. A polynomial p in n variables, $x_1,x_2,\dots x_n$ is symmetric if for any permutation $\sigma \in S_n$, $p(x_1,x_2,\dots x_n)=p(x_{\sigma (1)},x_{\sigma (2)},\dots ,x_{\sigma (n)})$. In [2] Wan proved that if $f(x)$ is exceptional over ${\mathbb{F}}_q$, then $f(x)$ is a permutation polynomial over ${\mathbb{F}}_q$ by way of p-adic lifting lemma and Newton’s formula about symmetric polynomials. Let $f\in {\mathbb{F}}_q[x]$ be a polynomial of degree n such that $1\le n\le q$ and let $s_k$ be the $k$th elementary symmetric polynomial of values $f(c)$, that is,

$$\prod _{c\in {\mathbb{F}}_q}(x-f(c))=\sum _{k=0}^q{(-1)}^k{s}_k{x}^{q-k}.$$

By studying elementary symmetric polynomials, Turnwald [3] proved a theorem giving nine characterizations of permutation polynomials. He also gave a very simple and elementary proof of Wan’s theorem ([2], Theorem 2.1) based on an elementary symmetric polynomial. Ensuring that a given polynomial is a permutation polynomial, there are several criteria with rather complicated conditions to satisfy. In [4], the additive character is used to prove $F(x)=G(x)+\gamma Tr(H(x))$ is permutation polynomial over $F_{2^n}$. In [5] complete permutation polynomials are constructed in four classes using additive character. In [6] evaluated class of the Weil sum using additive character. By applying some known results as in [7] and AGW Criterion [8], we have obtained several new types of permutation polynomials using the existence of almost one solution of $f(x)=d$ for any $d\in F_{q^n}$.

In [9,10,11,12,13], Yaun and Zeng investigated permutation polynomials in the form ${(x^{2^m}+x+\delta )}^s+x$ with different cases of m and s. Similarly in [10,14,15], Yaun and Zha developed polynomials ${(x^{p^m}-x+\delta )}^s+x$ which are over the field $F_{2^n}$. Jose et al. in [16] considered permutation polynomials with Carlitz rank 2 and developed a sharp lower bound for the weight of any PP polynomials. In [17] Vadiraja et al. investigated permutation polynomials over finite rings with respect to the sequences they generate. The sequences generated by various permutation polynomials are tested for randomness using well-known statistical methods. In [18] Sartaj et al. discussed c-differential uniformity and boomerang uniformity using additive characters of two classes of permutation polynomials. We have been inspired by [6] the method they have used and additive characters and its property to find Weil sum.

For $n\ge 1$, let $F_q[x_1,x_2,\dots ,x_n]$ be a ring of polynomial in n-indeterminates over $F_q$. In [19], they constructed permutation polynomial of n-indeterminates using additive and multiplicative characters over a finite ring. In [20], they discussed the relationship between planar functions and a permutation polynomial in n-indeterminates. Permutation polynomials have many applications in cryptography [21,22,23,24,25] coding theory [26,27,28] and combinatorial designs [29]. Permutations polynomials can be used as public-key cryptosystem, Rajesh et al. [30] proposed little dragon two cryptosystems by using permutation polynomials as a public key cryptosystem. In [23], Rajesh et al. proposed an efficient multivariate encryption scheme with the help of a group of permutation polynomials over finite fields, proposed cryptosystem was secure against usual known attacks on existing multivariate public-key cryptosystems. Khachatrian et al. [25] developed a new public-key encryption system based on permutation polynomials using permutation polynomials as a public polynomial for encryption.

In this paper, we have considered additive characters, multiplicative characters, Jacobi sum, and AGW Criterion to construct a new kind of permutation polynomials in one and more than one indeterminates and also proved affine polynomial is not a planar function. Quadratic forms have a major role to play in theory and methods of optimization. The criterion, recently discovered by Akbary et al. [8] (AGW) is a simple and effective method that establishes the permutation property of a mapping $F_{q^n}\to F_{q^n}$ through a commutative diagram. The main proofs in this paper completely depend on AGW Criterion and existence of at most one solution for the equation $f(x)=d$ for each $d\in F_q$, where $x\in F_q$. To have completeness in this section we discuss some of the results related to them. As a prerequisite to understanding and developing the theory, we need the existing definitions and results, which are listed here. In the following section, we made use of these concepts as tools for our work.

Lemma 1

([31]). The necessary and sufficient condition for a polynomial $f(x)\in F_q[x]$ to be a permutation of $F_q$ is any one of the following four conditions;

1. 

$f:c\to f(c)$ is surjective

2. 

$f:c\to f(c)$ is injective

3. 

The equation $f(x)=a$ has a solution in $F_q$ for each $a\in F_q$

4. 

For each $a\in F_q$, the equation $f(x)=a$ has a unique solution in $F_q$.

Lemma 2

(The AGW Criterion: [8]). If the maps g, $\overline{g}$, ω and $\overline{\omega }$ such that $g:K\to K$, $\overline{g}:L\to \overline{L}$, $\omega :K\to L$, and $\overline{\omega }:K$ $\to \overline{L}$ with $\overline{\omega }\circ g=\overline{g}\circ \omega$ and both ω and $\overline{\omega }$ surjective, where K, L and $\overline{L}$ be finite sets with $\left|L\right|=|\overline{L}|$ [Figure 1]. Then, the following statements are equivalent:

1. 

g is a bijection from K to K.

2. 

$\overline{g}$ is a bijection from L to $\overline{L}$ and g is injective on ${\omega }^{-1}(l)$ for each $l\in L$.

The major proofs in this work are based on the AGW Criterion, which simplifies the challenge of proving permutation polynomials over $F_{q^n}$ to that of proving permutation polynomials over its subfield $F_q$.

Trace functions are widely used in different types of concepts over finite fields. For properties of trace one can refer [31].

Definition 1

([31]). The trace $T{r}_{q^n/q}(\alpha )$ over $F_q$ for any $\alpha \in F_{q^n}$ is defined as $T{r}_{q^n/q}(\alpha )={\sum }_{i=0}^m{\alpha }^{q^i}$.

We use the following definition defined in [32].

Definition 2

([32]). Let $f(x)={\sum }_{i=0}^m{a}_i{x}^i\in F_{q^n}[x]$ with coefficients $a_i\in F_{q^n}$ then $T[f](x)=Tr(f(x))={\sum }_{i=0}^{m}Tr(a_i)x^i$.

Definition 3

([31]). The character function ${\chi }_1(c)=e^{\frac{2\pi iTr(c)}{p}}$ defined on additive group of $F_q$ is known as canonical additive character of $F_q$.

Definition 4

([31]). The function ${\chi }_b$ with ${\chi }_b(c)={\chi }_1(bc)$ for all $b,c\in F_{q^n}$ is known as additive character of $F_{q^n}$.

By setting, $b=0$ in above definition we get trivial additive character ${\chi }_0$, such that ${\chi }_0(c)=1$ for all $c\in F_{q^n}$.

Definition 5

([31]). A polynomial $L(x)$ with the coefficients in an extension field $F_{q^n}$ of the field $F_q$ represented as

$$L(x)=\sum _{i=0}^n{\alpha }_i{x}^{q^i}$$

is known as $q-$ linearized polynomial over $F_{q^n}$.

Definition 6

([31]). A polynomial in the form $A(x)=L(x)-\alpha$ where $L(x)$ is a linearized polynomial over $F_{q^n}$ and $\alpha \in F_{q^n}$, is an affine polynomial over $F_{q^n}$.

Definition 7

([33]). If for every $b\in F_q^{*}$, the function $f(x+b)-f(x)$ is a permutation of $F_q$, then $f:F_q\to F_q$ is known as a planar function.

The number of solutions of a polynomial function and properties of additive characters are interrelated to observe the permutation behavior of polynomials. We state the following results from [31], which we readily used in our work.

Theorem 1

([31]). A mapping $g:F_{q^n}\to F_{q^n}$ is a permutation polynomial if and only if for every $\alpha \in F_{q^n}^{*}$

$$\sum _{x\in F_{q^n}}{(-1)}^{Tr(\alpha g(x))}=0.$$

Theorem 2

([31]). If χ is a non-trivial additive character of the finite field $F_{q^n}$ then ${\sum }_{g\in F_{q^n}}^{}\chi (g)=0$.

Remark 1

([31]). For any $d\in F_{q^n}$

$$N=\frac{1}{q}\sum _{c\in F_{q^n}}^{}\sum _{\chi }\chi (f(c))\overline{\chi (d)}.$$

(1)

is the number of solutions of $f(x)=d$ in $F_{q^n}$.

Some of the required definitions and theorems to construct permutation polynomial in n-indeterminates are listed below.

Definition 8

([31]). A polynomial $f(x_1,\dots ,x_n)$ over $F_q$ is known as a permutation polynomial in n indeterminates, if for each $a\in F_q$, the equation $f(x_1,\dots ,x_n)=a$ has $q^{n-1}$ solutions in $F_q^n$.

Definition 9

([31]). Let $g\in F_q$ be a fixed primitive element. The function ${\psi }_j$ for each $j=0,1,\dots ,q-2$, is a multiplicative character of $F_q$ with

$${\psi }_j(g^k)=e^{2\pi ijk/(q-1)}fork=0,1,\dots ,q-2.$$

${\psi }_0$ is a trivial multiplicative character if ${\psi }_0(c)=1$ for all $c\in F_q^{*}$.

Definition 10

([31]). When q is odd, the quadratic character of $F_q$ is a real valued function η on $F_q^{*}$ with

$$\eta (c)=\left\{\begin{array}{cc}0\hfill & \mathit{if}c=0\hfill \\ 1\hfill & \mathit{if}\mathit{c}\mathit{is}\mathit{square}\mathit{of}\mathit{an}\mathit{element}\mathit{of}{F}_q^{*}\hfill \\ -1\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Definition 11

([31]). The function υ on $F_q$ is integer valued function if $\upsilon (b)=-1$ for $b\in F_q^{*}$ and $\upsilon (0)=q-1$.

Lemma 3

([31]). For any finite field $F_q$ we have

$$\sum _{c\in F_q}\upsilon (c)=0$$

(2)

and for any $b\in F_q$,

$$\sum _{c_1+c_2+\dots +c_n=b}\upsilon (c_1)\upsilon (c_2)\cdots \upsilon (c_k)=\left\{\begin{array}{cc}0\hfill & if1\le k\le m\hfill \\ \upsilon (b)q^{m-1}\hfill & ifk=m\hfill \end{array}\right.$$

(3)

where the sum is over all $c_1,c_2,\dots ,c_m\in F_q$ with $c_1+c_2+\dots c_m=b$.

Definition 12

([31]). Let ${\gamma }_1,\dots ,{\gamma }_k$ be k multiplicative characters of $F_q$. Then the sum

$$J({\gamma }_1,\dots ,{\gamma }_k)=\sum _{c_1+c_2+\dots +c_n=1}{\gamma }_1(c_1)\cdots {\gamma }_k(c_k),$$

(4)

with the summation extended over all $k-$ tuples $(c_1,\dots ,c_k)$ of elements of $F_q$ satisfying $c_1+c_2+\dots +c_n=1$, is called a Jacobi sum in $F_q$.

Definition 13

([31]). The homogeneous polynomial of degree 2, or the zero polynomial $f(x_1,\dots ,x_n)\in F_q[x_1,\dots ,x_n]$ is a quadratic form in n indeterminates over $F_q$ if,

$$f(x_1,\dots ,x_n)=\sum _{i,j=1}^n{a}_{ij}{x}_i{x}_{j}with{a}_{ij}=a_{ji}for1\le i\le j\le n.$$

Theorem 3

([31]). Every quadratic form over $F_q$, q odd, is equivalent to a diagonal quadratic form.

3. Construction of Permutation Polynomials Using Additive Character

In this section we construct the polynomials of the type $f(x)=a_t{x}^{q^t}+a_{t-1}{x}^{q^{t-1}}+\dots +a_1{x}^q+a_{0}x+a$ and $P(x)=f(Tr(x))+k(Tr(x))·T[h](x)$. We prove these two type of polynomials are permutation polynomials based on the additive character over $F_{q^n}$.

Theorem 4.

The affine polynomial $f(x)=a_t{x}^{q^t}+a_{t-1}{x}^{q^{t-1}}+\dots +a_1{x}^q+a_{0}x+a$ is a permutation polynomial over the finite field $F_{q^n}$ of characteristic p.

Proof. 

It is enough to prove that $f(x)=d$ has atmost one solution in $F_{q^n}$. We make use of additive character over $F_{q^n}$ and ${\sum }_{c\in F_{q^n}}{\chi }_b(f(c))={\sum }_{c\in F_{q^n}}{\chi }_b(L(c)+a)={\sum }_{c\in F_{q^n}}[{\chi }_b(a)+{\chi }_1(L(c))]={\chi }_b(a){\sum }_{c\in F_{q^n}}{\chi }_1(L(c))$ where $L(x)=b{a}_t{x}^{q^t}+b{a}_{t-1}{x}^{q^{t-1}}+\dots +b{a}_1{x}^q+b{a}_{0}x$ is a $q-$ polynomial over $F_{q^n}$.

Put $\tau (c)={\chi }_1(L(c)$ for all $c\in F_{q^n}$, by the property of $L(c)$ implies that $\tau (c)$ is an additive character of $F_{q^n}$. Thus,

$$\sum _{c\in F_{q^n}}{\chi }_1(L(c))=\sum _{c\in F_{q^n}}\tau (c)=\left\{\begin{array}{cc}q\hfill & \mathrm{if}\tau \mathrm{is}\mathrm{trivial}\hfill \\ 0\hfill & \mathrm{if}\tau \mathrm{is}\mathrm{nontrivial}.\hfill \end{array}\right.$$

From the above Remark 1,

$$\begin{array}{ll}N=\frac{1}{q}{\displaystyle \sum _{c\in F_{q^n}}^{}}{\displaystyle \sum _{{\chi }_b}}{\chi }_b(f(c))\overline{{\chi }_b(d)}& =\frac{1}{q}{\displaystyle \sum _{c\in F_{q^n}}^{}}{\displaystyle \sum _{{\chi }_b}}{\chi }_b(d){\chi }_1(L(c))\overline{{\chi }_b(d)}\\ & =\frac{1}{q}{\displaystyle \sum _{c\in F_{q^n}}}{\chi }_1(L(c))=\frac{1}{q}{\displaystyle \sum _{c\in F_{q^n}}}\tau (c)\end{array}$$

Hence $f(x)=d$ has atmost one solution in $F_{q^n}$. □

Example 1.

$a_2{x}^2+a_{1}x+a_0$is a permutation polynomial over$F_{q^n}$withqeven.

Proposition 1.

Let $F_{2^n}$ be a finite field of characteristic 2 and $f(x)=a_t{x}^{2^t}+a_{t-1}{x}^{2^{t-1}}+\dots +a_1{x}^2+a_{0}x+a$ be an affine polynomial over $F_{2^n}$. For $g(x)\in F_{2^n}[x]$ and $a\in F_{2^n}$, the polynomial $F(x)=f(x)+aTr(g(x))$ is a permutation polynomial over $F_{2^n}$ if and only if for all $\lambda \in F_{2^n}^{*}$,

$$\begin{array}{c}\hfill \sum _{x\in F_{2^n}}{(-1)}^{Tr(\lambda a)[1+Tr(g(x))]}=0\mathit{whenever}\sum _{i=0}^t{a}_i=0,\\ \hfill and\sum _{x\in F_{2^n}}{(-1)}^{Tr(\lambda a)[1+Tr(g(x))]+\lambda Tr(x)}=0\mathit{whenever}\sum _{i=0}^t{a}_i=1.\end{array}$$

Proof. 

If $L(x)$ is a linearised polynomial then $f(x)=L(x)+a$ is an affine polynomial over $F_{2^n}$ for any $a\in F_{2^n}$ and $tr(f(x))=Tr(a)+(a_0+a_1+\dots +a_t)Tr(x)$. Hence, $Tr(\lambda F(x))=Tr(\lambda a)[1+Tr(g(x))]+\lambda (a_0+a_1+\dots +a_t)Tr(x)$.

By Theorem 1 $F(x)$ is a permutation polynomial if and only if ${\sum }_{x\in F_{2^n}}{(-1)}^{Tr(\lambda F(x))}=0$ for all $\lambda \in F_{2^n}^{*}$.

Combining these facts together we can write,

$$\sum _{x\in F_{2^n}}{(-1)}^{Tr\lambda F(x)}=\left\{\begin{array}{cc}{\sum }_{x\in F_{2^n}}{(-1)}^{Tr(\lambda a)[1+Tr(g(x))]}=0\hfill & \mathrm{if}{\sum }_{i=0}^n{a}_i=0\hfill \\ {\sum }_{x\in F_{2^n}}{(-1)}^{Tr(\lambda a)[1+Tr(g(x))]+\lambda Tr(x)}=0\hfill & \mathrm{if}{\sum }_{i=0}^n{a}_i=1\hfill \end{array}\right.$$

which completes the proof. □

Thus the affine polynomial $f(x)=a_t{x}^{q^t}+a_{t-1}{x}^{q^{t-1}}+\dots +a_1{x}^q+a_{0}x+a$ over $F_{q^n}$ is a good candidate to be a permutation polynomial. But using the Definition 7, we observe that the same function is not a planar function, which is shown in the following result.

Theorem 5.

An affine polynomial $f(x)=a_t{x}^{q^t}+a_{t-1}{x}^{q^{t-1}}+\dots +a_1{x}^q+a_{0}x+a$ over $F_{q^n}$ is not planar function over $F_{q^n}$.

Proof. 

The function $f(x)$ is planar if $f(x+b)-f(x)$ permutes $F_{q^n}$ for all $b\in F_{q^n}^{*}$. Here we need to show that $f(x+b)-f(x)$ does not permute $F_{q^n}$

In fact, $f(x+b)-f(x)$

$$\begin{array}{cc}\hfill & =a_r{(x+b)}^{q^t}+a_{r-1}{(x+b)}^{q^{t-1}}+\dots +a_1{(x+b)}^q+a_0(x+b)+a-\hfill \\ \hfill & (a_t{x}^{q^t}+a_{t-1}{x}^{q^{t-1}}+\dots +a_1{x}^q+a_{0}x+a)\hfill \end{array}$$

As $F_{q^n}$ is a field of characteristic p, this expression reduces to

$$a_r{b}^{q^t}+a_{r-1}{b}^{q^{t-1}}+\dots +a_{0}b=L(b),$$

which is a constant, which shows that $f(x+b)-f(x)$ cannot be a permutation of the field. □

Reis and Wang [32] defined $T[f](x)$ as shown in Definition 2 with the help of a polynomial $f(x)\in F_{q^n}[x]$ using Trace function. Here we construct another type of permutation polynomial with the help of $T[f](x)$.

Theorem 6.

Let $P(x)=f(Tr(x))+k(Tr(x))·T[h](x)$ where $f(x),k(x)\in F_{q^n}[x]$, and $T[h](x)\in F_q[x]$ with $h(x)\in F_q[x]$ and $k(Tr(F_{q^n}))\subseteq F_q^{*}$. Then $P(x)$ is a permutation polynomial if $Tr(\lambda )=0$ for all $\lambda \in F_q^{*}$.

Proof. 

By AGW Criterion $P(x)$ is a permutation polynomial over $F_{q^n}$ if and only if $Q(x)=T[f](x)+k(x)·T[h](x)$ is a permutation polynomial over $F_q$. The given statement is symbolized using the diagram Figure 2.

From the Theorem 1 for each $\lambda \in F_q^{*}$,

$$\begin{array}{ll}{\sum }_{x\in F_q}{(-1)}^{Tr(\lambda Q(x))}=& {\sum }_{x\in F_q}{(-1)}^{Tr(\lambda )T[f](x)+Tr(\lambda )k(x)·T[h](x)}\\ & ={\displaystyle \sum _{x\in F_q}}{(-1)}^{Tr(\lambda )\{T[f](x)+k(x)T[h](x)\}}\end{array}$$

This sum is zero if $Tr(\lambda )=0$, which completes the proof. □

Corollary 1.

Let $P(x)\in F_{q^n}[x]$ and m be an odd positive integer such that $P(x)=f(Tr(x))+T[h](x)$, where $f(x)\in F_{q^n}[x]$ is of degree m and $T[h](x)\in F_q[x]$ with $h(x)\in F_q[x]$. If $Tr(f(Tr(y)))=0,$ for all $y\in F_{q^n}$ then $P(x)$ is a permutation polynomial over $F_{q^n}$.

Proof. 

By taking $k(x)=1$ in the previous theorem, $Q(x)$ reduces to $Q(x)=T[f](z)+T[h](z)=T[h](z)$, for each $z\in Tr(F_{q^n})$. By AGW Criterion $P(x)$ is permutation polynomial if and only if $Q(x)$ is permutation polynomial. Here $Q(x)$ being a $p-$ polynomial will be bijective when m is odd. □

4. Permutation Polynomials in $\mathit{n}$-Indeterminates

In this section, we construct permutation polynomials in n-indeterminates using multiplicative character, Jacobi sum, and quadratic form-related properties.

Theorem 7.

Let $F_q$ be a finite field of characteristic p. Let $f(x_1,\dots ,x_n)$ be a quadratic form over $F_q$, with n indeterminates and odd q. Then $f(x_1,\dots ,x_n)$ is a permutation polynomial over $F_q^n$ if at least one of the coefficients $a_i$ of the corresponding diagonal quadratic form is zero.

Proof. 

Being a quadratic form $f(x_1,\dots ,x_n)$ is always equivalent to the diagonal quadratic form $a_1{x}_1^2+\dots +a_n{x}_n^2$. Thus it is enough to show that $a_1{x}_1^2+\dots +a_n{x}_n^2=b$ has exactly $q^{n-1}$ solutions for each $b\in F_q$.

Using trivial multiplicative character ${\psi }_0$ and quadratic character $\eta$, with $c_1,c_2,\dots ,c_n\in F_q^n$, the number of solutions of $a_1{x}_1^2+\dots +a_n{x}_n^2$ can be computed as

$$\begin{array}{cc}\hfill & N(a_1{x}_1^2+\dots +a_n{x}_n^2)\hfill \\ \hfill & =\sum _{c_1+c_2+\dots +c_n=b}N(a_1{x}_1^2=c_1)N(a_2{x}_2^2=c_2)\cdots N(a_n{x}_n^2=c_n)\hfill \\ \hfill & =\sum _{c_1+c_2+\dots +c_n=b}[1+\eta (c_1{a}_1^{-1})][1+\eta (c_2{a}_2^{-1})]\cdots [1+\eta (c_n{a}_n^{-1})]\hfill \\ \hfill & =\sum _{c_1+c_2+\dots +c_n=b}[{\psi }_0(c_1{a}_1+\eta (c_1{a}_1))][{\psi }_0(c_2{a}_2)+\eta (c_2{a}_2)]\cdots [{\psi }_0(c_n{a}_n)+\eta (c_n{a}_n)]\hfill \\ \hfill & ={\psi }_0(a_1)\cdots {\psi }_0(a_n)J_b({\psi }_0,\dots ,{\psi }_0)+\eta (a_1,\dots ,(a_n))J_b(\eta ,\dots ,\eta )\hfill \\ \hfill & =q^{n-1}+\eta (a_1\cdots a_n)J_b(\eta ,\dots ,\eta )\hfill \end{array}$$

(5)

If n is even and $b\ne 0$,

$$J(\eta ,\dots ,\eta )=-q^{(\frac{n-2}{2})}\eta [{(-1)}^{\frac{n}{2}}]$$

so that

$$N=q^{n-1}-q^{(\frac{n-2}{2})}\eta [{(-1)}^{\frac{n}{2}}{a}_1\cdots a_n]$$

If n is odd and $b\ne 0$,

$$J(\eta ,\dots ,\eta )=-q^{(\frac{n-1}{2})}\eta [{(-1)}^{\frac{n-1}{2}}]$$

so that

$$N=q^{n-1}-q^{(\frac{n-1}{2})}\eta [{(-1)}^{\frac{n-1}{2}}b{a}_1\cdots a_n].$$

In both of the above cases if ${\prod }_{i=1}^n{a}_i$ is zero, the number of solutions will be $q^{n-1}$ for each $b\in F_q^{*}$.

When $b=0$,

$$J(\eta ,\dots ,\eta )=-q^{(\frac{n-2}{2})}\eta [{(-1)}^{\frac{n}{2}}]$$

and hence the number of solutions will be $q^{n-1}$. □

Corollary 2.

For odd q, the quadratic form f over $F_q$ is permutation polynomial over $F_q^2$ if $a_1{a}_2=0$.

Proof. 

Let $f(x_1,x_2)$ be a quadratic form over $F_q$. It is equivalent to $a_1{x}_1^2+a_2{x}_2^2=b$ for $b\in F_q$.

With $c_1,c_2\in F_q$ we obtain,

$$\begin{array}{cc}\hfill & N(a_1{x}_1^2+a_2{x}_2^2=b)=\sum _{c_1+c_2=b}N(a_1{x}_1^2=c_1)N(a_2{x}_2^2=c_2)\hfill \\ \hfill =& \sum _{c_1+c_2=b}[1+\eta (c_1{a}_1^{-1})][1+\eta (c_2{a}_2^{-1})]=q+\eta (a_1{a}_2)J_b(\eta ,\eta ).\hfill \end{array}$$

If $a_1{a}_2=0$ then $\eta (a_1{a}_2)=0$ and hence, $N=q$. □

Theorem 8.

Let $F_q$ be a finite field with q even. For odd n

$$x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+a{x}_n^2$$

is a permutation polynomial over $F_q^n$.

Proof. 

It is enough to show that

$$N(x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+a{x}_n^2=b)$$

has $q^{n-1}$ solutions over $F_q^n$ for each $b\in F_q$.

$$\begin{array}{cc}\hfill & N(x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+a{x}_n^2=b)\hfill \\ \hfill & =\sum _{c_1+c_2=b}N(x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}=c_1)N(a{x}_n^2=c_2)\hfill \end{array}$$

(6)

$N(x_1{x}_2=b)$ is $q-1$ if $b\ne 0$ and $2q-1$ if $b=0$. These two cases can be combined as $N(x_1{x}_2=b)=q+\upsilon (b)$. So we have, $N(x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}=c_1)=q^{n-2}+\upsilon (c_1)q^{\frac{n-3}{2}}$. And by using the property of diagonal quadratic form,

$N(a{x}_n^2=c_2)=N(x_n^2=a^{-1}{c}_2)=\lambda (a^{-1}{c}_2)=\lambda (a^{-1})J_b(\lambda )=q^{1-1}=1$. Hence from the Equation (6),

$$N=\sum _{c_1+c_2=b}(q^{n-2}+\upsilon (c_1)q^{\frac{n-3}{2}})·1=q^{n-1}+q^{\frac{n-3}{2}}\sum _{c_1+c_2=b}\upsilon (c_2)=q^{n-1}.$$

(7)

 □

Theorem 9.

For odd n and even q, the quadratic form

$$x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+x_n^2$$

is a permutation polynomial over $F_q^n$.

Proof. 

It is enough to show that

$$N(x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+x_n^2=b)$$

has $q^{n-1}$ solutions over $F_q^n$ for each $b\in F_q$.

Let f be a quadratic form equivalent to

$$f(x_1,x_2,\dots ,x_n)=\sum _{1\le i\le j\le n}{a}_{ij}{x}_i{x}_j$$

Suitable non-singular linear substitution for indeterminates will be the same in the cases of either some $a_{ii}$ being zero or all $a_{ii}$ being zero.

Without loss of generality assuming $a_{12}\ne 0$, the non singular linear substitution will be

$$\begin{array}{cc}\hfill & x_2=a_{12}^{-1}(y_2+a_{13}{y}_3+\dots +a_{1n}{y}_n)\hfill \\ \hfill & x_i=y_{i}fori\ne 2\hfill \end{array}$$

Hence by assigning arbitrary values to $x_1,x_2,\dots ,x_{n-1}$ and the corresponding value of $x_n$,

$$N(x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+x_n^2=b)=q^{n-1}.$$

 □

The following result is to prove that $f(x_1,x_2,\dots ,x_n)=x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+x_n^2$ is not a planar function using nontrivial additive character of $F_q$.

Theorem 10.

Let $F_q$ be a finite field with q even. For odd n

$$x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+x_n^2$$

is not a planar function over $F_q^n$.

Proof. 

Let $f(x_1,x_2,\dots ,x_n)=x_1{x}_2+x_3{x}_4+\dots +x_{n-2}{x}_{n-1}+x_n^2$.

Then for $(a_1,a_2,\dots ,a_n)\in F_q^n$,

$$f(x_1+a_1,x_2+a_2,\dots ,x_n+a_n)-f(x_1,x_2,\dots ,x_n)=a_1{a}_2+a_3{a}_4+\dots +a_n^2,$$

which is a constant. This shows that $f(x_1,x_2,\dots ,x_n)$ cannot be a planar function. □

5. Conclusions

Permutation of finite field elements is a useful property that increases the richness of finite fields. The construction of such polynomials has become a major part of research in recent years. We contributed a notable work of construction of Permutation polynomials using additive characters, multiplicative characters, and Trace functions for the polynomials in one and more than one indeterminates. The AGW criterion is effectively used to understand the permutation behavior of the constructed polynomials. Linearised polynomials and associated affine polynomials can be suitably modified to form permutation polynomials. The construction of planar functions may also be an interesting work using a similar method in the future.