Permutational behavior of reversed Dickson polynomials over finite fields

In this paper, we use the method developed previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,k}(1,x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$. Our results extend the known results from the case $0\le k\le 3$ to the general $k\ge 0$ case.


Introduction
Let F q be the finite field of characteristic p with q elements. Associated to any integer n ≥ 0 and a parameter a ∈ F q , the n-th Dickson polynomials of the first kind and of the second kind, denoted by D n (x, a) and E n (x, a), are defined for n ≥ 1 by ] means the largest integer no more than n 2 . In 2012, Wang and Yucas [7] further defined the n-th Dickson polynomial of the (k + 1)-th kind D n,k (x, a) ∈ F q [x] for n ≥ 1 by Hou, Mullen, Sellers and Yucas [5] introduced the definition of the reversed Dickson polynomial of the first kind, denoted by D n (a, x), as follows if n ≥ 1 and D 0 (a, x) = 2. To extend the definition of reversed Dickson polynomials, Wang and Yucas [7] defined the n-th reversed Dickson polynomial of (k + 1)-th kind D n,k (a, x) ∈ F q [x], which is defined for n ≥ 1 by and D 0,k (a, x) = 2 − k.
It is well known that D n (x, 0) is a permutation polynomial of F q if and only if gcd(n, q− 1) = 1, and if a = 0, then D n (x, a) induces a permutation of F q if and only if gcd(n, q 2 − 1) = 1. Besides, there are lots of published results on permutational properties of Dickson polynomial E n (x, a) of the second kind (see, for example, [2]). In [7], Wang and Yucas investigated the permutational properties of Dickson polynomial D n,2 (x, 1) of the third kind. They got some necessary conditions for D n,2 (x, 1) to be a permutation polynomial of F q .
Hou, Mullen, Sellers and Yucas [5] considered the permutational behavior of reversed Dickson polynomial D n (a, x) of the first kind. Actually, they showed that D n (a, x) is closely related to almost perfect nonlinear functions, and obtained some families of permutation polynomials from the revered Dickson polynomials of the first kind. In [4], Hou and Ly found several necessary conditions for the revered Dickson Polynomials D n (1, x) of the first kind to be a permutation polynomial. Recently, Hong, Qin and Zhao [3] studied the revered Dickson polynomial E n (a, x) of the second kind that is defined for n ≥ 1 by and E 0 (a, x) = 1. In fact, they gave some necessary conditions for the revered Dickson polynomial E n (1, x) of the second kind to be a permutation polynomial of F q . Regarding the revered Dickson polynomial D n,2 (a, x) ∈ F q [x] of the third kind, from its definition one can derive that for each x ∈ F q . Using (1.2), one can deduce immediately from [3] the similar results on the permutational behavior of the reversed Dickson polynomial D n,2 (a, x) of the third kind. On the other hand, by using the method presented by Hong, Qin and Zhao in [3], Cheng, Hong and Qin [1] obtained the results on the permutational behavior of the reversed Dickson polynomial D n,3 (a, x) of the fourth kind. In this paper, our main goal is to continue to use the method developed by Hong, Qin and Zhao in [3] to investigate the reversed Dickson polynomial D n,k (a, x) of the (k +1)-th kind which is defined by (1.1) if n ≥ 1 and D 0,k (a, x) := 2 − k. For a = 0, we write x = y(a − y) with an indeterminate y = a 2 . Then one can rewrite D n,k (a, x) as We have D n,k a, a 2 4 = (kn − k + 2)a n 2 n . (1.4) In fact, (1.3) and (1.4) follow from Theorem 2.2 (i) and Theorem 2.4 (i) below. It is easy to see that if char(F q ) = 2, then D n,k (a, x) = E n (a, x) if k is odd and D n,k (a, x) = D n (a, x) if k is even. We also find that D n,k (a, x) = D n,k+p (a, x), so we can restrict p > k. Thus we always assume p = char(F q ) > 3 in what follows. The paper is organized as follows. First in section 2, we study the properties of the reversed Dickson polynomial D n,k (a, x) of the fourth kind. Subsequently, in Section 3, we prove a necessary condition for the reversed Dickson polynomial D n,k (1, x) of the k +1-th kind to be a permutation polynomial of F q and then introduce an auxiliary polynomial to present a characterization for D n,k (1, x) to be a permutation of F q . From the Hermite criterion [6] one knows that a function f : F q → F q is a permutation polynomial of F q if and only if the i-th moment Thus to understand well the permutational behavior of the reversed Dickson polynomial D n,k (1, x) of the fourth kind, we would like to know if the i-th moment a∈Fq D n,k (1, a) i is computable. We are able to treat with this sum when i = 1. The final section is devoted to the computation of the first moment a∈Fq D n,k (1, a).

Reversed Dickson polynomials of the k + 1-th kind
In this section, we study the properties of the reversed Dickson polynomials D n,k+1 (a, x) of the fourth kind. Clearly, if a = 0, then if n is even. Therefore, D n,k+1 (0, x) is a PP (permutation polynomial) of F q if and only if n is an even integer with gcd( n 2 , q − 1) = 1. In what follows, we always let a ∈ F * q . First, we give a basic fact as follows.
. Then f (x) is a PP of F q if and only if cf (dx) is a PP of F q for any given c, d ∈ F * q . Then we can deduce the following result.
Theorem 2.2. Let a, b ∈ F * q . Then the following are true. (i). One has D n,k (a, x) = a n b n D n,k (b, b 2 a 2 x). (ii). We have that D n,k (a, x) is a PP of F q if and only if D n,k (1, x) is a PP of F q .
Proof. (i). By the definition of D n,k (a, x), we have as required. Part (i) is proved.
(ii). Taking b = 1 in part (i), we have D n,k (a, x) = a n D n,k 1, It then follows from Lemma 2.1 that D n,k (a, x) is a PP of F q if and only if D n,k (1, x) is a PP of F q . This completes the proof of part (ii). So Theorem 2.2 is proved.
Theorem 2.2 tells us that to study the permutational behavior of D n,k (a, x) over F q , one only needs to consider that of D n,k (1, x). In the following, we supply several basic properties on the reversed Dickson polynomial D n,k (1, x) of the fourth kind. The following result is given in [3] and [5] without proof. For its proof, one can see [1].
Each of the following is true.
(i). For any integer n ≥ 0, we have If n 1 and n 2 are positive integers such that n 1 ≡ n 2 (mod q 2 − 1), then one has . the first identity is true for the cases that n = 0 and 1. Now let n ≥ 2. Then one has D n,k 1, kn − k + 2 2 n as desired. So the first identity is proved. Now we turn our attention to the second identity. Let x = 1 4 , then there exists . So by the definition of the n-th reversed Dickson polynomial of the k + 1-th kind, one has Proof. We consider the following two cases.
Case 2. x = 1 4 . Then by Theorem 2.4 (i), we have This concludes the proof of Proposition 2.5.
By Proposition 2.5, we can obtain the generating function of the reversed Dickson polynomial D n,k (1, x) of the k + 1-th kind as follows.
Proposition 2.6. The generating function of D n,k (1, x) is given by Proof. By the recursion presented in Proposition 2.5, we have Thus the desired result follows immediately.
Now we can use Theorem 2.4 to present an explicit formula for D n,k (1, x) when n is a power of the characteristic p. Then we show that D n,k (1, x) is not a PP of F q in this case.
Proposition 2.7. Let p = char(F q ) > 3 and s be a positive integer. Then Proof. We consider the following two cases. Case 1. x = 1 4 . For this case, putting x = y(1 − y) in Theorem 2.4 (i) gives us that as desired. Case 2. x = 1 4 . By Theorem 2.4 (i), one has 2 p s D p s ,k 1, as required. So Proposition 2.7 is proved.
It is well known that every linear polynomial over F q is a PP of F q and that the monomial x n is a PP of F q if and only if gcd(n, q − 1) = 1. Then by Proposition 2.7, we have the following result.
The latter one is impossible since p−1 2 | gcd p s −1 2 , q − 1 implies that Thus D p s ,k (1, x) is not a PP of F q . Proposition 2.9. Let p = char(F q ) > 3 and s and l be integers such that 0 < s < ℓ. Then Proof. We consider the following two cases. Case 1. x = 1 4 . For this case, putting x = y(1 − y) in Theorem 2.4 (i) gives us that where u = 2y − 1 and u 2 = 1 − 4x. So we obtain that as desired. Case 2. x = 1 4 . By Theorem 2.4 (i), one has D p s +p ℓ ,k 1, Thus the required result follows. So Proposition 2.9 is proved.
Then we obtain a characterization for D n,k (1, x) to be a PP of F q as follows.
Theorem 2.11. Let q = p e with p > 3 being a prime and e being a positive integer. Let Proof. First, we show the sufficiency part. Let f be 2-to-1 and f (y) = kn−k+2 To show that D n,k (1, x) is a PP of F q , it suffices to show that x 1 = x 2 that will be done in what follows.
The proof of Theorem 2.11 is complete.

3.
A necessary condition for D n,k (1, x) to be permutational and an auxiliary polynomial In this section, we study a necessary condition on n for D n,k (1, x) to be a PP of F q . In particular, if k = 3, then it is easy to check that for n ≥ 0 , then one can easily show that the sequences {D n,k (1, 1)|n ∈ N} are periodic with the smallest positive periods 6. In fact, one has Theorem 3.1. Assume that D n,k (1, x) is a PP of F q with q = p e and p > 3. Then n ≡ 1 (mod 6).
Proof. Let D n,k (1, x) be a PP of F q . Then D n,k (1, 0) and D n,k (1, 1) are distinct. Then by the above results, the desired result n ≡ 1 (mod 6) follows immediately.
Let n, k be nonnegative integers. We define the following auxiliary polynomial p n,k (x) ∈ Z[x] by x j for n ≥ 1 and p 0,k (x) := 2 n (2−k). Then we have the following relation between D n,k (1, x) and p n,k (x). First, let x ∈ F q \ { 1 4 }. Then there exists y ∈ F q 2 \ { 1 2 } such that x = y(1 − y). Let u = 2y − 1. It then follows from Theorem 2.4 (i) that as desired. So (3.1) holds in this case. Consequently, we let x = 1 4 . Then by Theorem 2.4 (i), we have D n,k 1, On the other hand, we can easily check that p n,k (0) = kn − k + 2. Therefore 4. The first moment a∈Fq D n,k (1, a) In this section, we compute the first moment a∈Fq D n,k (1, a). By Proposition 2.6, one has Moreover, by Theorem 2.4 (ii), it follows that for any t n+ℓ(q 2 −1) (4.2) Then (4.1) together with (4.2) gives that for any x = 1 4 , we have where Lemma 4.1.
For convenience, let a n := a∈Fq D n,k (1, a).