#### Abstract

Let denote the finite field with elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

#### 1. Introduction

For a prime power , let denote the finite field of order , and the multiplicative group. A polynomial is called a permutation polynomial (PP) over , if the associated polynomial mapping is a permutation of . They have applications in coding theory, cryptography, and combinatorial design theory [1, 2]. Thus, in both theoretical and applied aspects, finding new PPs is of great interest. Permutation polynomials with few terms attract many authors’ attention for their simple algebraic structures. In particular, there are many results about permutation binomials and trinomials [3–5].

Permutation polynomials attract peoples’ interest for their extraordinary properties and algebraic forms. Complete permutation polynomial (CPP) is a permutation polynomial such that is also a permutation polynomial. Mann introduced CPPs in the construction of orthogonal Latin squares [6]. Orthomorphisms map each maximal subgroup of the additive group of half into itself and half into its complement, and they have a single fixed point and are the same as CPPs in even characteristic. Nonlinear orthomorphisms (or CPPs) are of cryptographic interest, and Mittenthal used them for the design of a nonlinear dynamic substitution device [7, 8]. PPs have been applied in the Lay–Massey scheme, the block cipher SMS4, the stream cipher Loiss [1, 9, 10], the design of Hash functions, quasigroups, and also in the constructions of some cryptographically strong functions [2, 11–14]. In [15–17], the authors investigated the set stability of switched delayed logical networks (SDLNs), the optimal state estimation of finite-field networks (FFNs), and containment problem of finite-field networks (FFNs).

A monomial permutes if , and they are the simplest kind of permutation polynomials. For binomials and trinomials, the permutation properties are not so easy to determine. Carlitz studied permutation binomials in 1962 [3]. In [18], Carlitz and Wells found that for large enough than , the polynomial might be a permutation polynomial over . Hou and Lappano studied permutation binomials of the forms and [19]. However, only a limited number of constructions are known for PPs. More recent constructions of PPs can be found in [4, 20–31], and the references therein.

For two positive integers and with , denotes the trace function from to , that is,

Permutation polynomials of the form have been studied for special , with even characteristic [32, 33], and for the case , i.e., permutation trinomials [4, 34]. Later, Zheng, Yuan, and Yu investigated permutation polynomials with the formover , where is a positive integer, and [35]. For this, they used the AGW criterion to prove three classes of permutation trinomials. For the fourth class of permutation trinomials, a symbolic computation method related with resultants was used. And, they found a new relation with a class of PPs of the formover , where is arbitrary and . This class of permutation polynomials are related to . Based on their relationship, the aforementioned class of permutation trinomials without restriction on is derived.

In the study of CPPs, Li et al., found that certain polynomials over can have the same permutation properties as over for positive divisor of such that is odd [36]. They constructed some permutation binomials, and thus more permutation polynomials of the form can be obtained, and here are constants. They also studied permutation polynomials over of the form , and Niho-type permutation trinomials over were constructed.

In this paper, some preliminaries are presented first in Section 2. In Sections 3 and 4, we construct some new classes of PPs including the following:(a)In [35], permutation polynomials of the kind are proposed, but with all coefficients 1, here we add a variable to the coefficient to investigate more general situations. In Proposition 1, PPs of the kind are studied.(b)In [36], permutation behavior of is studied for odd value , and here in Proposition 10, all positive values of will be investigated.(c)In [37], Xu et al. proposed a class of permutation polynomials of the form for satisfying . In Proposition 11, the case of is studied.(d)In Proposition 13, a totally new class of PPs is proposed.(e)And the PPs in the other propositions are presented in these two sections.

In [37], Xu et al. proposed two classes of permutation polynomials of the form and over and , respectively, and sufficient conditions were given. In this work, we investigate the necessities of the above two classes of permutation polynomials, and the necessary conditions are given in Section 5.

#### 2. Preliminaries

Let be the finite field which is an extension of of degree and denotes its multiplicative group. Use to denote the norm function from to , i.e., for , . Let be a linearized polynomial, with . Define

There is the following result about the number of solutions of over .

Lemma 1 (see [38]). *Let with . Then has 1 root in if and only if .*

Lemma 2 (see [39]). *Let , where is a positive integer. The quadratic equation , where and , has roots in if and only if .*

Lemma 3 (see [40]). *For a positive integer , satisfying . Then the quadratic equation has*(i)*both solutions in the unit circle if and only if and ,*(ii)*exactly one solution in the unit circle if and only if and .*

Lemma 4 (see [35]). *Let be integers with and . Let and . Then permutes for each if and only if permutes .*

Lemma 5 (see [41]). * Let with , and let . Then permutes if and only if the following two conditions hold:*(i)*,*(ii)* permutes where denotes the -th root of unity in .**Using the same idea as in [[45], Proposition 3], we can get the following lemma.*

Lemma 6 (see [24]). *Let be integers with and . Let and . Then permutes for each if and only if permutes .*

For each element in the finite field , define . The unit circle of is the setwhich is also denoted by occasionally.

The following lemma can be verified without much difficulty.

Lemma 7 (see [24]). *Each nonzero element in the finite fields has a unique expression of the following form:with and .*

Lemma 8 (see [24]). *Let be a positive integer, and . Then the equation the following equation:*(i)*has at most one possible solution , when ,*(ii)*has solutions, when and ,*(iii)*has no solutions, when and .*

Proof. Let be a solution, then

Taking the power on both sides of the above equation,

Substituting equation (8) into the above equation,

Taking the power of the above equation,

Substituting equation (8) into the above equation,

Noting that , the above equation can be transformed into

For case (ii), we need to show that there are solutions. Taking the power of the right side of equation (13),

Then we can find that is a solution of equation (8). Let us consider the equation

Since and , the above equation has solutions. For every such , we can find that is a solution of (2).

#### 3. Permutation Polynomials That Can be Transformed to the Case of Monomials

In this section, we study the permutation behavior of six kinds of polynomials. Propositions 1 and 2 have the property that after some operations the complicated terms can be canceled. Propositions 3–6 have the property that exponents can be simplified. Denote .

Proposition 1. *Let be even with , and . Thenand it is a permutation polynomial over for satisfying .*

Proof. It can be found that from the assumption . Let us rewrite the polynomial in the following:

Set , equation (83) becomes

By Lemma 5, is a permutation polynomial if and only ifpermutes . On the set , and become

It is necessary to show thathas no zeros on . The above equation can be written as

We can find that . Set , then equation (22) becomes

For Lemma 3, we have which lies in . And equation (23) has no solutions in the unit circle by our assumption.

Let us make the following transformations for on the unit circle :

Thus, is a permutation of .

*Example 1. *Set . Using Magma, it can be verified thatis a permutation polynomial over for .

Proposition 2. *Let be positive integers with and , and satisfying . Then the polynomialis a permutation polynomial over .*

Proof. Let and .

It can be verified that the above diagram is commutative. By assumption, we have for . So, maps to . We need to check that is bijective on . For , it is not difficult to show thatwhich means that . Thuspermutes since .

For satisfying , the following holds

Ifthen . Since , we have . That is is injective on the set for . Using the AGW criterion [42], is a permutation over .

*Example 2. *Set . Using Magma, it can be verified that the following polynomial:is a permutation polynomial over for .

In the following, Propositions 3 and 4 are concerned with degree two extensions, but with different exponents and coefficients. Propositions 5 and 6 are concerned with degree three extensions.

Proposition 3. *Le be positive integers with , and satisfying . Then the polynomialis a permutation over .*

Proof. For Lemma 5, we have . Since , is a permutation polynomial if and only ifpermutes the unit circle . We claim thathas no zeros in the unit circle. Otherwise, take the power on both sides of the above equation

From the above two equations, we havewhich is equivalent to

After simplification, the above equation can be transformed into

Thus

By Lemma 3, the above equation has no zeros in the unit circle since .

Now, equation (33) becomeswhich permutes since .

*Example 3. *Set . Let be a primitive element of and . Then the polynomialis a permutation polynomial over when .

Proposition 4. *Let be positive integers with , and satisfying . Then the polynomialis a permutation polynomial over .*

Proof. For Lemma 5, we have . Since , is a permutation polynomial if and only ifpermutes , which means thathas no zeros in . Squaring both sides of the above equationwhich is equivalent to

By Lemma 2, the above equation has no solutions in .

*Example 4. *Set . Let be a primitive element of , and . Using Magma, it can be verified thatis a permutation polynomial when .

Proposition 5. *Let be positive integers, and . Then the polynomialis a permutation polynomial for the following two cases:*(i)* and ,*(ii)* and .**Here .*

Proof. For Lemma 5, we have . Since , then is a permutation polynomial over if and only ifpermutes . We only need to show that is nonzero on the set , which can be deduced from Lemma 8.

*Example 5. *Let . Using Magma, it can be verified that the following polynomialis a permutation polynomial over when and .

Proposition 6. *Let be positive integers satisfying . Then the polynomialis a permutation polynomial over , where , and . Here is defined as in (1).*

Proof. For Lemma 5, we have . Since , is a permutation polynomial over if and only ifpermutes , which is equivalent towhich has no solutions in .

Since , the above equation has a solution in . But , and lies in if and only . This is contradiction with the assumption that .

By Lemma 1, equation (53) is in fact a permutation polynomial over . It has only one solution , which does not belong to the set .

*Example 6. *Set . Then for all values in , using Magma, it can be verified thatis a permutation polynomial over .

#### 4. Construction of Permutation Polynomials with Two or More terms

##### 4.1. Three Classes of Permutation Polynomials of Degree Three or Four Extensions over

In the following, Propositions 7–9 are concerned with PPs over field extensions of degrees 3 and 4.

Proposition 7. *Let , . Then is a class of permutation polynomials of if one of the following conditions holds:*(i)*, and is a polynomial over ,*(ii)*, and is a positive integer.*

Proof. We only prove case (ii), and case (i) can be proved similarly.

Due to Lemma 4, is a permutation polynomial if and only ifis a permutation of . But

That is , which is a permutation of for .

*Remark 1. *For to be a permutation polynomial over , satisfies . Here we give two explicit expressions of .

Proposition 8. *Let , . Then is a permutation polynomial if one of the following conditions holds:*(i)*, where is a polynomial over and satisfying ,*(ii)*, where is a positive integer and satisfying .*

Proof. We only consider case (i), and case (ii) can be proved in a similar way.

Due to Lemma 6, is a permutation polynomial if and only ifis a permutation of . But

That is , which is a permutation of for .

*Remark 2. *Note that in the above proposition for odd characteristic.

Proposition 9. *For a positive integer , a fixed , and , the polynomialis a permutation of .*

Proof. [[43], Proposition 4] says thatis a permutation of . Set with . Then we have

Since is a permutation for every , setting ,is a permutation of for every . Let .is a permutation of for every .

Note that, in Proposition 9, is unconnected with Combining with Lemma 4, the following result is obtained.

Corollary 1For a positive integer and , the polynomialis a permutation of .

*Example 7. *Set . Using Magma, it can be verified that the following polynomial:is a permutation over for .

##### 4.2. PPs of Type over

The following proposition is concerned with PPs of both odd and even positive values of .

Proposition 10. *Let be a positive integer, , and is a primitive element of . Then is a permutation polynomial of if and only ifwhere and .*

Proof. We can find thatand . Let us consider the nonzero elements in .

It is not difficult to check thatwithfor . And we havefor . Then maps to the setfor . For fixed , the elements in are different. Then is a permutation polynomial if and only iffor , which is equivalent to thatfor and . After simplification, we get the result.

*Example 8. *Let , and be a primitive element of . For , the polynomialis not a permutation polynomial when , and here .

##### 4.3. PPs of Type over

In the following proposition, we study PPs of type over with constant .

Proposition 11. *For positive integers with . For any , the polynomialis a permutation of , where satisfying .*

Proof. We prove that has at most one solution for any , which is equivalent towhich has a unique solution.

It can be verified that for when . Let , then . Equation (76) can be rewritten aswhich is equivalent to

That is,

So,by the assumption. Now, means that is a permutation of . Therefore there is a unique satisfying equation (80).

*Example 9. *Let , then . Let be any element, , satisfying . Using Magma, it can be verified thatis a permutation polynomial over .

##### 4.4. PPs of Type over

In the following proposition, we consider PPs of type over .

Proposition 12. *Let be positive integers satisfying , where is an odd integer. Let , then the polynomialis a permutation of , with .*

Proof. Since , and for odd,

To prove that is a permutation polynomial, it is enough to prove that for any , has a unique solution. That isis satisfied by at most one . By (83), taking the power on both sides of the above equation gives the equivalent equation:

First, if there exists a solution such thatthen , for the right side of equation (85) is also zero. In this case, the above equation becomes

Second, let us assume that . Since taking the power, the left side of equation (85) is 1, and the right side is in the unit circle , that is,for some . But since ,for some . Thus

And equation (85) can be rewritten as

Since , the left side of the above equation becomes

So, we have .

Now, the above two situations can be summarized. For every element , if satisfies equation (87), there are two possibilities for the values of as considered above. But is not the solution. For substituting it into equation (85), the left side becomes

It is not equal to the right side which now becomes 1. If does not satisfy equation (87), and if is a solution of equation (85), then . The second situation tells us that the only solution is .

*Example 10. *Set . Let be any element, and . Using Magma, it can be verified thatis a permutation polynomial over .

##### 4.5. PPs of Type over

In the following proposition, we consider PPs of type over .

Proposition 13. *For the finite field , let be in the unit circle, and satisfying . Then the linearized polynomialis a permutation polynomial of .*

Proof. By the assumption, it can be checked that

Otherwise from , we have , contradiction with the condition that .

Since is a linearized polynomial; to verify that it is a permutation polynomial, it is necessary to check thathas only the zero solution. There are two situations to be considered.

First assume that is a solution of (97), and then

That is,

If , the above equation becomes , contradiction. So, let us assume that , then

But we have , that is, , and thus

So,which implies thatcontradiction with equation (96).

Second let us assume that ; by Lemma 7, we can writewith and . Substituting the above into equation (97),

That is,

If . The above equation becomes , contradiction. So, ; that is,

Then from equation (106),