1 Further results on some classes of permutation polynomials over finite fields Xiaogang Liu Abstract Let Fq denote the finite field with q elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with their studies, and get some further results about the permutation properties of the permutation polynomials. Also, some new classes of permutation polynomials are constructed. For these, we alter the coefficients, exponents or the underlying fields, etc. Index Terms Finite field; Permutation polynomial; AGW criterion; Trace function I. INTRODUCTION F F∗ For a prime power q, let q denote the finite field of order q, and q the multiplicative group. A polynomial f(x) ∈ Fq is called a permutation polynomial (PP) over Fq, if the associated polynomial mapping f : c → f(c) is a permutation of Fq. In recent years, permutation polynomials have become an interesting area of research. They have applications in coding theroy, cryptography and combinatorial design theory [6, 19]. 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 [2, 5, 13, 18]. For recent achievements on the study of permutation polynomials, the reader can consult [5, 8, 13, 15–17, 21], and the references therein.. Fqn k F Permutation polynomials of the form x+ q (x ) have been studied for special γ ∈ qn , with even characteristic arXiv:1907.03386v1 [cs.IT] 8 Jul 2019 [3, 4], and for the case n =2, i.e., trinomial permutations [9, 13]. Later, Zheng, Yuan and Yu investigated permutation polynomials with the form cx − xs + xqs over Fqn , where s is a positive integer, and c ∈ Fq [22]. For this, they used the well-known result (Lemma 3) to prove three classes of permutation trinomials. For the fourth class of permutation trinomials, symbolic computation method related with resultants was used. And, they found a new relation with a class of PPs of the form k (xq − x + δ)s + cx This work is supported in part by ”Funding for scientific research start-up” of Nanjing Tech University. X. Liu is with College of Computer Science and Technology, Nanjing Tech University, Nanjing City, Jiangsu Province, PR China 211800 e-mail:[email protected]. 2 over Fqn , where δ is arbitrary and c ∈ Fq. This class of permutation polynomials are related to δ. Based on their relationship, the aforementioned class of pemutation trinomials without restriction on δ are derived. Complete permutation polynomial (CPP) is a permutation polynomial such that f(x)+ x is also a permutation polynomial. CPPs were introduced with the construction of Latin squares [10]. In the study of CPPs, Li etc., k found that certain polynomials over F2n can have the same permutation properties as ax + bx over F2m for n positive divisor m of n such that m is odd [11]. They constructed some permutation binomials, and thus more n k n F∗ permutation polynomials of the form a[Trm(x)] +u(c+x)(Trm(x)+x)+bx can be obtained, here a,b,c,u ∈ 2m F pm pm are constants. They also studied permutation polynomials over p2m of the form ax + bx + h(x ± x), and F Niho-type permutation trinomials over p2m were constructed. In this paper, we revisit the permutation polynomials proposed in [11, 22]. Based on their results we make some modifications on the condtions, and get some further results which are generalizations. Also, we construct some new classes of permutation polynomials in Sections II and III. Before going on, let us present the following lemmas, which might be useful for our study. ∗ F F m Lemma 1: [22] Let m, k be integers with 0 <k<m and l = gcd(k,m). Let c ∈ ql and g(x) ∈ q [x]. Then qk qk g(x − x + δ)+ cx permutes Fqm for each δ ∈ Fqm if and only if h(x)= g(x) − g(x)+ cx permutes Fqm . Using the same idea as in [22, Proposition 3], we can get the following lemma. ∗ F F m Lemma 2: Let m, k be integers with 0 <k<m and l = gcd(k,m). Let c ∈ ql and g(x) ∈ q [x]. Then qk qk g(x + x + δ)+ cx permutes Fqm for each δ ∈ Fqm if and only if h(x)= g(x) + g(x)+ cx permutes Fqm . r (q−1)/d Lemma 3: [23] Let d,r > 0 with d | q − 1, and let h(x) ∈ Fq[x]. Then f(x) = x h(x ) permutes Fq if and only if the following two conditions hold: (i) gcd(r, (q − 1)/d) = 1; r (q−1)/d (ii) x h(x) permutes µd, where µd denotes the d-th root of unity in Fq. F∗ n b Lemma 4: [20] For a positive integer m, and a,b ∈ 22m satisfying Tr1 ( a2 )=0. Then the quadratic equation x2 + ax + b =0 has (i) both solutions in the unit circle if and only if a m b m 1 b = a2m and Tr1 ( a2 )= Tr1 ( a2m+1 )=1; (ii) exactly on solution in the unit circle if and only if a 2m+1 2m+1 2m+1 2 2m 2m+1 b 6= a2m and (1 + b )(1 + a + b )+ a b + a b =0. II. PERMUTATION POLYNOMIALS WITH TWO OR THREE TERMS In this section, we study the permutation behavior of four kinds of polynomials. Two kinds of them have three terms, or to some extent are related with a three term polynomial, see Corollary 1 and Proposition 2. Two kinds of them have two terms, see Proposition 3 and Proposition 4. They come from transformations from permutation polynomials proposed in [11, 22]. F F∗ Proposition 1: For a positive integer m, a fixed δ ∈ 23m and c ∈ 2m , the polynomial m 2m f(x) = (x2 + x + δ)2 +1 + cx 3 F is a permutation of q3m . Proof: [12, Proposition 4] says that m 2m g(x) = (x2 + x + δ)2 +1 + x F F∗ is a permutation of q3m . Set x = c0y, with c0 ∈ 2m . Then we have 2m 22m+1 g(x)= g(c0y) = ((c0y) + c0y + δ) + c0y m 2m = c2(y2 + y + 1 δ)2 +1 + c y 0 c0 0 m 2m = c2((y2 + y + 1 δ)2 +1 + 1 y). 0 c0 c0 1 Since g(x) is a permutation for every δ ∈ F 3m , setting δ0 = δ, q c0 2m 22m+1 1 f0(y) = (y + y + δ0) + y c0 1 F m F m is a permutation of q3 for every δ0 ∈ q3 . Let c0 = c , 2m 22m+1 f(y) = (y + y + δ0) + cy F F is a permutation of q3m for every δ0 ∈ q3m . Note that in Proposition 1, c is unconnected with δ. Combing with Lemma 1, the following result follows. F∗ Corollary 1: For a positive integer m and c ∈ 2m , the polynomial m 2m 2m g(x)= x2 (2 +1) + x2 +1 + cx F is a permutation of 23m . Example 1: Set m =3. Using Magma, it can be verified that the following polynomial g(x)= x520 + x65 + cx F F∗ is a permutation over 512 for c ∈ 8. In Theorem 10 [22], permutation polynomials of the kind x + xs + xqs are proposed, but with all coefficients 1, q+1 here we add a variable to the coefficient to investigate more general situations. Denote µq+1 = {x|x =1}. m q2+q+1 Proposition 2: Let q =2 be even with q ≡ 1 (mod 3), and s = 3 . Then f(x)= cx + xs + cqxqs F F∗ m q+1 is a permutation polynomial over q2 for c ∈ q2 satisfying Tr1 (c )=0. Proof: It can be found that 3|(q + 2) from the assumption q ≡ 1 (mod 3). Let us rewrite the polynomial f(x) in the following q+2 (q−1)+1 q ( q(q+2) +1)(q−1)+1 f(x)= cx + x 3 + c x 3 . (1) q+2 Set u = 3 , equation (1) becomes − − f(x)= x(c + x(q 1)u + cqx(qu+1)(q 1)). By Lemma 3, f(x) is a permutation polynomial if and only if − g(x)= x(c + xu + cqxqu+1)q 1 4 3u q+2 permutes µq+1. On the set µq+1, x = x = x and g(x) becomes − − g(x)= x(c + xu + cqx u+1)q 1. It is necessary to show that − c + xu + cqx u+1 =0 has no zeros on µq+1. The above equation can be written as c + xu + cqx2u =0. (2) q+2 u We can find that gcd(u, q +1)= gcd( 3 , q +1)=1. Set z = x , then equation (2) becomes 1 c z2 + z + =0. (3) cq cq b q+1 F For Lemma 4, we have a2 = c which lies in q. And equation (3) has no solutions in the unit circle by our assumption. Let us make the following transformations for g(x) on the unit circle µq+1, − − cq +x u+cx 1+u g(x) = x c+xu+cq x−u+1 − cq x+x1 u+cxu = c+xu+cq x−u+1 cq x3u+x2u+cxu = c+xu+cqx2u u cq x2u+xu+c = x c+xu+cqx2u = xu. Thus, g(x) is a permutation of µq+1. Example 2: Set m =4. Using Magma, it can be verified that f(x)= cx + x91 + c16x1456 F 4 17 is a permutation polynomial over 256 for Tr1(c )=0.