Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Let Fq denote the finite field with q elements. Permutation polynomials over finite fields have important applications inmany areas of science and engineering such as coding theory, cryptography, and combinatorial design.)e 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.


Introduction
For a prime power q, let F q denote the finite field of order q, and F * q the multiplicative group. A polynomial f(x) ∈ F q is called a permutation polynomial (PP) over F q , if the associated polynomial mapping f: c ⟶ f(c) is a permutation of F q . ey have applications in coding theory, cryptography, and combinatorial design theory [1,2]. us, 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][4][5].
Permutation polynomials attract peoples' interest for their extraordinary properties and algebraic forms. Complete permutation polynomial (CPP) is a permutation polynomial such that f(x) + x 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 F q 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][12][13][14]. In [15][16][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).
For two positive integers m and n with m|n, Tr n m (·) denotes the trace function from F p n to F p m , that is, Tr n m (x) � x + x p m + x p 2m + · · · + x p (n/m−1)m .
Permutation polynomials of the form x + cTr n m (x k ) have been studied for special c ∈ F q n , with even characteristic [32,33], and for the case n � 2, i.e., permutation trinomials [4,34]. Later, Zheng, Yuan, and Yu investigated permutation polynomials with the form cx − x s + x qs , (2) over F q n , where s is a positive integer, and c ∈ F q [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 form over F q n , where δ is arbitrary and c ∈ F q . is class of permutation polynomials are related to δ. Based on their relationship, the aforementioned class of permutation trinomials without restriction on δ is derived.
In the study of CPPs, Li et al., found that certain polynomials over F 2 n can have the same permutation properties as ax k + bx over F 2 m for positive divisor m of n such that n/m is odd [36]. ey constructed some permutation binomials, and thus more permutation polynomials of the form a[Tr n m (x)] k + u(c + x)(Tr n m (x) + x) + bx can be obtained, and here a, b, c, u ∈ F * 2 m are constants. ey also studied permutation polynomials over F p 2m of the form ax p m + bx + h(x p m ± x), and Niho-type permutation trinomials over F p 2m 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 x + x s + x qs 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 cx + x s + c q x qs are studied. (b) In [36], permutation behavior of x 2 2m − 1/3+1 + bx is studied for odd value m, and here in Proposition 10, all positive values of m will be investigated. (c) In [37], Xu et al. proposed a class of permutation polynomials of the form ( In Proposition 11, the case of (2 m + 2)(−s) ≡ 2 m − 1(mod2 2m − 1) is studied. (d) In Proposition 13, a totally new class of PPs 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 ( + bx over F 2 2m and F 2 2k , 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.

Preliminaries
Let F q n be the finite field which is an extension of F q of degree n and F * q n denotes its multiplicative group. Use N to denote the norm function from F q n to F q , i.e., for x ∈ F q n , ere is the following result about the number of solutions of L(x) over F q 3 .
Lemma 2 (see [39]). Let q � 2 k , where k is a positive integer. e quadratic equation Lemma 3 (see [40]). For a positive integer m, a, b ∈ F * Lemma 4 (see [35]). Let m, k be integers with 0 < k < m and l � gcd(k, m). Let c ∈ F * q l and g( Lemma 5 (see [41]). Let d, r > 0 with d|q − 1, and let only if the following two conditions hold: Using the same idea as in [ [45], Proposition 3], we can get the following lemma.
Lemma 6 (see [24]). Let m, k be integers with 0 < k < m and l � gcd(k, m). Let c ∈ F * q l and g( For each element x in the finite field F 2 2m , define x � x 2 m . e unit circle of F 2 2m is the set which is also denoted by μ q+1 occasionally.
2 Mathematical Problems in Engineering e following lemma can be verified without much difficulty.
Lemma 7 (see [24]). Each nonzero element x in the finite fields x has a unique expression of the following form: with u ∈ F * 2 m and λ ∈ U.
Lemma 8 (see [24]). Let m be a positive integer, and a, b ∈ F * 2 3m . en the equation the following equation: (i) has at most one possible solution Proof. Let x be a solution, then Taking the 2 m power on both sides of the above equation, Substituting equation (8) into the above equation, Taking the 2 m power of the above equation, Substituting equation (8) into the above equation, Noting that x 2 3m � x, the above equation can be transformed into For case (ii), we need to show that there are 2 m solutions. Taking the 2 m power of the right side of equation (13), en we can find that x � a 2 2m b 2 m is a solution of equation (8). Let us consider the equation Since a 2 2m +2 m +1 � 1 and 2 3m − 1 � (2 m − 1) (2 2m + 2 m + 1), the above equation has 2 m − 1 solutions. For every such c, we can find that c + a 2 2m b 2 m is a solution of (2).

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 μ q+1 � x|x q+1 � 1 .
and it is a permutation polynomial over Proof. It can be found that 3|(q + 2) from the assumption q ≡ 1(mod3). Let us rewrite the polynomial f(x) in the following: Set

By Lemma 5, f(x) is a permutation polynomial if and only if
It is necessary to show that has no zeros on μ q+1 . e above equation can be written as We can find that gcd(u, q + 1) � gcd(q + 2/3, q + 1) � 1. Set z � x u , then equation (22) becomes For Lemma 3, we have b/a 2 � c q+1 which lies in F q . And equation (23) has no solutions in the unit circle by our assumption.
Let us make the following transformations for g(x) on the unit circle μ q+1 :

Mathematical Problems in Engineering
us, g(x) is a permutation of μ q+1 .
. It can be verified that the above diagram is commutative. By assumption, we have x + b ≠ 0 for x ∈ S. So, g(x) maps S to S. We need to check that g(x) is bijective on S. For x ∈ S, it is not difficult to show that which means that , the following holds If Using the AGW criterion [42], f(x) is a permutation over Using Magma, it can be verified that the following polynomial: is a permutation polynomial over F 64 for b 3 ≠ 1, b 9 � 1. 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.
permutes the unit circle μ 2 m +1 . We claim that has no zeros in the unit circle. Otherwise, take the 2 m power on both sides of the above equation From the above two equations, we have which is equivalent to After simplification, the above equation can be transformed into us By Lemma 3, the above equation has no zeros in the unit circle since Tr m 1 (a 2 m +1 δ 2 m +1 /(a 2 m +1 + δ 2 m +1 + 1) 2 ) � 0. Now, equation (33) becomes which permutes μ 2 m +1 since gcd(r, 2 2m − 1) � 1.
is a permutation polynomial over F 2 2m .
permutes F * 2 m , which means that has no zeros in F * 2 m . Squaring both sides of the above equation which is equivalent to By Lemma 2, the above equation has no solutions in F 2 m .

Proposition 5.
Let r, m, s be positive integers, and a, b ∈ F * 2 3m . en the polynomial is a permutation polynomial for the following two cases: Here gcd(r, 2 3m − 1) � 1.
Using Magma, it can be verified that the following polynomial is a permutation polynomial over F 2 9 when a 73 ≠ 1 and (a 64 + a 72 + 1/a 73 ) 73 ≠ 1.

Proposition 6.
Let r, s, m be positive integers satisfying Here u is defined as in (1).
is is contradiction with the assumption that a + b + δ + 1 ≠ 0.
By Lemma 1, equation (53) is in fact a permutation polynomial over F 2 3m . It has only one solution x 0 , which does not belong to the set μ 2 2m +2 m +1 .

Construction of Permutation
Polynomials with Two or More terms

ree Classes of Permutation Polynomials of Degree ree or Four
Extensions over F q . In the following, Propositions 7-9 are concerned with PPs over field extensions of degrees 3 and 4.

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Proposition 7. Let c ∈ F * q , δ ∈ F q 3 . en f(x) � g(x q − x + δ) + cx is a class of permutation polynomials of F q 3 if one of the following conditions holds: and u(x) is a polynomial over F q 3 , (ii) g(x) � x i(q 2 +q+1) , and i is a positive integer.
Proof. We only prove case (ii), and case (i) can be proved similarly.
Due to Lemma 4, f(x) is a permutation polynomial if and only if is a permutation of F q 3 . But Here we give two explicit expressions of g(x).
is a permutation polynomial if one of the following conditions holds: Proof. We only consider case (i), and case (ii) can be proved in a similar way.

Due to Lemma 6, f(x) is a permutation polynomial if and only if
is a permutation of F q 4 . But at is h(x) � cx, which is a permutation of F q 4 for c ∈ F * q .
Remark 2. Note that in the above proposition c 0 ∉ F * q for odd characteristic.

Proposition 9.
For a positive integer m, a fixed δ ∈ F 2 3m , and c ∈ F * 2 m , the polynomial

Proof. [[43], Proposition 4] says that
is a permutation of F q 3m . Set x � c 0 y, with c 0 ∈ F * 2 m . en we have Since g(x) is a permutation for every δ ∈ F q 3m , setting δ 0 � 1/c 0 δ, is a permutation of F q 3m for every δ 0 ∈ F q 3m . Note that, in Proposition 9, c is unconnected with δ. Combining with Lemma 4, the following result is obtained.
Corollary 1For a positive integer m and c ∈ F * 2 m , the polynomial is a permutation of F 2 3m .
Example 7. Set m � 3. Using Magma, it can be verified that the following polynomial: is a permutation over F 512 for c ∈ F * 8 .

PPs of Type
e following proposition is concerned with PPs of both odd and even positive values of m.
Proof. We can find that with for i � 0, 1, 2. And we have for i ≠ j. en g(x) maps U i to the set for i � 0, 1, 2. For fixed i, the elements in V i are different. en g(x) is a permutation polynomial if and only if for i ≠ j, which is equivalent to that for i ≠ i ′ and 1 ≤ k, k ′ ≤ 2 2m − 1/3. After simplification, we get the result.

PPs of Type
In the following proposition, we study PPs of type (bx + δ) 2 m +1 + x 2 m + cx over F 2 km with constant c.
Proposition 11. For positive integers m, n, k with n � km, 2|k. For any δ ∈ F 2 n , the polynomial Proof. We prove that g(x) � d has at most one solution for any d ∈ F 2 n , which is equivalent to which has a unique solution.

PPs of Type
In the following proposition, we consider PPs of type (x 2 m + x + δ) − s + bx over F 2 2m .

Proposition 12. Let s, m be positive integers satisfying
, where m is an odd integer. Let δ ∈ F 2 m , then the polynomial Proof. Since gcd(2 m + 2, 2 m + 1) � 1, and gcd(2 m + 2, 2 m − 1) � gcd(3, 2 m − 1) � 1 for m odd, To prove that g(x) is a permutation polynomial, it is enough to prove that for any d ∈ F 2 2m , g(x) � d has a unique solution. at is is satisfied by at most one x. By (83), taking the (2 m + 2) th power on both sides of the above equation gives the equivalent equation: First, if there exists a solution x such that Mathematical Problems in Engineering then x � d/b, for the right side of equation (85) is also zero. In this case, the above equation becomes Second, let us assume that x 2 m + x + δ ≠ 0. Since taking the (2 m + 1) th power, the left side of equation (85) is 1, and the right side is in the unit circle U, that is, for some λ 0 ∈ U. But since gcd(2 m + 2, 2 m + 1) � 1, for some λ ∈ U. us And equation (85) can be rewritten as Since δ 2 m � δ, the left side of the above equation becomes It is not equal to the right side which now becomes 1. If d does not satisfy equation (87), and if x is a solution of equation (85), then x 2 m + x + δ ≠ 0. e second situation tells us that the only solution is Example 10. Set m � 3, s � 6. Let δ ∈ F 2 3 be any element, and b ∈ F 2 3 \F 2 . Using Magma, it can be verified that is a permutation polynomial over F 2 6 .

PPs of Type
In the following proposition, we consider PPs of type Proposition 13. For the finite field F 2 2m , let b ′ ∈ U be in the unit circle, and b ∉ F 2 m satisfying b 2(2 m − 1) b ′ 3 � 1. en the linearized polynomial is a permutation polynomial of F 2 2m .
Proof. By the assumption, it can be checked that Since g(x) is a linearized polynomial; to verify that it is a permutation polynomial, it is necessary to check that has only the zero solution. ere are two situations to be considered. First assume that x ∈ F * 2 m is a solution of (97), and then at is, If b ′ � 1, the above equation becomes b � 0, contradiction. So, let us assume that b ′ ≠ 1, then contradiction with equation (96). Second let us assume that x ∈ F 2 2m F 2 m ; by Lemma 7, we can write with u ∈ F * 2 m and λ ∈ U. Substituting the above x into equation (97), at is, . e above equation becomes bλ � 0, contradiction. So, λ 4 ≠ 1/b ′ ; that is, 8 Mathematical Problems in Engineering en from equation (106), (108) Since u ∈ F * 2 m , we have that u 2 m � u. e above equation becomes at is, which can be rewritten as Let λ 0 � λ 2 ; equation (111) can be transformed into Take derivative of the above equation; Substituting is a double root of (21), which has three roots at most counting multiplicity. But, contradiction with equation (107). e third root of (112) is So And equation (106) becomes by assumption, contradiction.
Example 11. Set m � 4. Let b ′ ∈ U be any element of the unit circle in F 2 8 Magma, it can be verified that is a permutation polynomial over F 2 8 .

4.6.
PPs of Type x r (x q− 1 + a) over F q e . In the following proposition, we study PPs of type x r (x q− 1 + a) over F q e .
Proof. is can be obtained using Lemma 5.

Necessities of Two Kinds of Permutation Polynomials
In this section, we investigate the necessities of two classes of permutation polynomials studied in [37], where the sufficient conditions are given. + bx over F 2 2m , and sufficient conditions were given. In the following proposition, we consider its necessary conditions. Proposition 15. For a positive integer m and a fixed δ ∈ F 2 2m with Tr 2m m (δ) ≠ 0, let

PPs of Type
where b ∈ F 2 2m . When b ∉ F 2 m , g(x) is permutation polynomial if and only if b + b m � b 2 m +1 .
Proof. As pointed out at the beginning of this subsection, ( [42], Proposition 10) gives the sufficiency verification. Now let us consider the necessity.
Assume that g(x) is a permutation polynomial. en for every d ∈ F 2 2m , g(x) � d has a unique solution. at is, Mathematical Problems in Engineering has at most one possible root in F 2 2m . Squaring both sides of the above equation, we get the following equivalent equation at is, which can be transformed into which implies that has a unique solution in F 2 2m . en for x 1 ≠ x 2 ∈ F 2 2m with x 1 a solution of equation (127), the following equation: cannot hold. Adding the above two equations, does not hold for any x 2 different from x 1 . Now let y � x 1 + x 2 . With x 1 fixed and x 2 varying, y can be any nonzero element of the finite field F 2 2m . So, y 2 m+1 + δ + δ 2 m y 2 m + b 2 + 1 y 2 + δ + δ 2 m y � 0 (130) has only the solution zero in F 2 2m , and this is from the assumption that g(x) is a permutation polynomial. If a nonzero solution y ∈ F 2 2m of equation (130) exists. Taking the 2 m th power, y 2 + δ + δ 2 m y + b 2 m+1 + 1 y 2 m+1 + δ + δ 2 m y 2 m � 0.
(LCNs), stability analysis of nonlinear systems on time scales is further investigated in [45], the H ∞ problem of nonlinear descriptor systems (NDSs) is investigated in [46], and the containment problem of finite-field networks (FFNs) with fixed topology (FT-FFNs) and switching topology (ST-FFNs) is studied in [17]. In these works, the authors utilized algebraic properties of some state space representations, and containment problem of finite-field networks (FFNs) is studied using semitensor product (STP) of matrices. By considering their permutation properties, some further results might be obtained.

Data Availability
e data used to support the theoretical findings were generated using Magma.

Disclosure
Reference [24] is just an incomplete draft of this work.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.