DICKSON POLYNOMIALS OVER FINITE FIELDS QIANG WANG AND JOSEPH L. YUCAS Abstract. In this paper we introduce the notion of Dickson polynomials of the (k + 1)-th kind over finite fields Fpm and study basic properties of this family of polynomials. In particular, we study the factorization and the per- mutation behavior of Dickson polynomials of the third kind. 1. Introduction m Let Fq be a finite field of q = p elements. For any integer n ≥ 1 and a th parameter a in a field Fq, we recall that the n Dickson polynomial of the first kind Dn(x, a) ∈ Fq[x] is defined by bn/2c X n n − i D (x, a) = (−a)ixn−2i. n n − i i i=0 th Similarly, the n Dickson polynomial of the second kind En(x, a) ∈ Fq[x] is defined by bn/2c X n − i E (x, a) = (−a)ixn−2i. n i i=0 For a 6= 0, we write x = y + a/y with y 6= 0 an indeterminate. Then Dickson polynomials can often be rewritten (also referred as functional expression) as a an D (x, a) = D y + , a = yn + , n n y yn and a yn+1 − an+1/yn+1 E (x, a) = E y + , a = , n n y y − a/y √ √ √ √ n √ for y 6= 0, ± a; For y = ± a, we have En(2 a, a) = (n+1)( a) and En(−2 a, a) = √ n (n + 1)(− a) . It is well known that Dn(x, a) = xDn−1(x, a) − aDn−2(x, a) and En(x, a) = xEn−1(x, a) − aEn−2(x, a) for any n ≥ 2. In the case a = 1, we denote the nth Dickson polynomials of the first kind and the second kind by Dn(x) and En(x) respectively. It is well known that these Dick- son polynomials are closely related to Chebyshev polynomials by the connections Dn(2x) = 2Tn(x) and En(2x) = Un(x), where Tn(x) and Un(x) are Chebyshev polynomials of degree n of the first kind and the second kind, respectively. More information on Dickson polynomials can be found in [8]. In the context of complex functions, Dickson polynomials of other kinds have already been introduced, see for example [6, 9]. In the context of finite fields, some properties such as recursive Key words and phrases. finite fields, Dickson polynomials, permutation polynomials, factorization. Research of Qiang Wang is partially supported by NSERC of Canada. 1 2 QIANG WANG AND JOSEPH L. YUCAS relations remain the same but the emphases are rather different. For example, per- mutation property over finite fields is one of properties which have attract a lot of attention due to their applications in cryptography. In this paper, we study Dick- son polynomials of the higher kinds over finite fields. For any k < p and constant th a ∈ Fq, we define n Dickson polynomials Dn,k(x, a) of the (k + 1)-th kind and th the n reversed Dickson polynomials Dn,k(x, a) of the (k + 1)-th kind in Section 2. Moreover, we give the relation between Dickson polynomials of the (k + 1)-th kind and Dickson polynomials of the first two kinds, the recurrence relation of Dickson polynomials of the (k + 1)-th kind in terms of degrees for a fixed k and its gen- erating function, functional expressions, as well as differential recurrence relations. Some general results on functional expression reduction and permutation behavior of Dn,k(x, a) are also obtained in Section 2. Then we focus on Dickson polynomials of the third kind. In Section 3, we show the relation between Dickson polynomials of the third kind and Dickson polynomials of the second kind and thus obtain the factorization of these polynomials. Finally, we study the permutation behavior of Dickson polynomials of the third kind Dn,2(x, 1) in Section 4. Our work is moti- vated by the study of Dickson polynomials of the second kind given by Cipu and Cohen (separately and together) in [3, 4, 5], which aimed to address conjectures of existence of nontrivial Dickson permutation polynomials of second kind other than several interesting exceptions when the characteristics is 3 or 5. We obtain some necessary conditions for Dn,2(x, 1) to be a permutation polynomial (PP) of any fi- nite fields Fq. We also completely describe Dickson permutation polynomials of the third kind over any prime field following the strategy of using Hermite’s criterion and Gr¨obner basis over rings started in [3, 4, 5]. 2. Dickson polynomial of the (k + 1)-th kind th Definition 2.1. For a ∈ Fq, and any positive integers n and k, we define the n Dickson polynomial of the (k + 1)-th kind Dn,k(x, a) over Fq by bn/2c X n − ki n − i D (x, a) = (−a)ixn−2i. n,k n − i i i=0 th Definition 2.2. For a ∈ Fq, and any positive integers n and k, we define the n reversed Dickson polynomial of the (k + 1)-th kind Dn,k(a, x) over Fq by bn/2c X n − ki n − i D (a, x) = (−1)ian−2ixi. n,k n − i i i=0 Remark 2.3. For n = 0, we define Dn,k(x, a) = 2 − k = Dn,k(a, x). It is easy to see that Dn,0(x, a) = Dn(x, a) and Dn,1(x, a) = En(x, a). Moreover, we can have the following simple relation (2.1) Dn,k(x, a) = kDn,1(x, a) − (k − 1)Dn,0(x, a) = kEn(x, a) − (k − 1)Dn(x, a). It is easy to see that if char(Fq) = 2, then Dn,k(x, a) = Dn(x, a) if k is even and Dn,k(x, a) = En(x, a) if k is odd. So we can assume char(Fq) is odd and we can also restrict k < p because Dn,k+p(x, a) = Dn,k(x, a). Remark 2.4. The fundamental functional equation is y2n + kay2n−2 + ··· + kan−1y2 + an D (y + ay−1, a) = n,k yn DICKSON POLYNOMIALS OVER FINITE FIELDS 3 y2n + an ka y2n − an−1y2 √ = + , for y 6= 0, ± a, yn yn y2 − a √ √ n where Dn,k(±2 a, a) = (± a) (kn − k + 2). Remark 2.5. For a fixed k and any n ≥ 2, we have the following recursion: Dn,k(x, a) = xDn−1,k(x, a) − aDn−2,k(x, a), where D0,k(x, a) = 2 − k and D1,k(x, a) = x. Using this recursion, we can obtain the generating function of these Dickson polynomials. Lemma 2.6. The generating function of Dn,k(x, a) is ∞ X 2 − k + (k − 1)xz D (x, a)zn = . n,k 1 − xz + az2 n=0 Proof. ∞ 2 X n 1 − xz + az Dn,k(x, a)z n=0 ∞ ∞ ∞ X n X n+1 X n+2 = Dn,k(x, a)z − x Dn,k(x, a)z + a Dn,k(x, a)z n=0 n=0 n=0 ∞ X n = 2 − k + xz − (2 − k)xz + (Dn+2,k(x, a) − xDn+1,k(x, a) + aDn,k(x, a)) z n=0 = 2 − k + (k − 1)xz. Stoll [9] has studied these Dickson-type polynomials with coefficients over C. We note that in our case all the coefficients of Dn,k(x, a) are integers. Hence Lemma 17 in [9] can be modified to the following. Lemma 2.7. The Dickson polynomial Dn,k(x, a) satisfies the following difference equation 4 2 2 00 3 0 2 (A4x +aA2x +a A0)Dn,k(x, a)+(B3x +aB1x)Dn,k(x, a)−(C2x +aC0)Dn,k(x, a) = 0, where A4,A2,A0,B3,B1,C2,C0 ∈ Z satisfy A4 = B3 = n(1 − k) 2 A2 = −(n − 1)(2 − k) − 2(2n + 1)(2 − k) + 4n 2 A0 = 4(n − 1)(2 − k) + 8(2 − k) 2 B1 = −3(n − 1)(2 − k) + 2(4n − 3)(2 − k) − 8n 3 C2 = n (1 − k) 2 C0 = −n(n − 1)(n − 2)(2 − k) − 2n(3n − 4)(2 − k) − 8n. In particular, for k = 0, 1, 2, we have 2 00 0 2 (x − 4a)Dn,0(x, a) + xDn,0(x, a) − n Dn,0(x, a) = 0, 2 00 0 (x − 4a)Dn,1(x, a) + 3xDn,1(x, a) − n(n + 2)Dn,1(x, a) = 0, 2 2 00 2 0 2 2 x (x − 4a)Dn,2(x, a) + x(x + 8a)Dn,2(x, a) − (n x + 8a)Dn,2(x, a) = 0, 4 QIANG WANG AND JOSEPH L. YUCAS ∗ Theorem 2.8. Suppose ab is a square in Fq . Then Dn,k(x, a) is a PP of Fq if and only if Dn,k(x, b) is a PP of Fq. Furthermore, p n p Dn,k(α, a) = ( a/b) Dn,k(( b/a)α, b). Proof. bn/2c n X n − ki n − i pa/b D (pb/aα, b) = (pa/b)n (−b)i(pb/aα)n−2i n,k n − i i i=0 bn/2c X n − ki n − i = (pa/b)n (−b)i(pb/aα)n−2i n − i i i=0 bn/2c X n − ki n − i = (−a)i(α)n−2i n − i i i=0 = Dn,k(α, a). So we can focus on a = ±1. When a = 1, we denote by bn/2c X n − ki n − i D (x) = (−1)ixn−2i.