Finite Fields and Their Applications

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1 Finite Fields and Their pplications ) Contents lists available at ScienceDirect Finite Fields and Their pplications Permutation polynomials over finite fields from a powerful lemma Pingzhi Yuan a,1, Cunsheng Ding b, a School of Mathematics, South China Normal University, Guangzhou , PR China b Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong article info abstract rticle history: Received 15 December 2010 Revised 25 March 2011 ccepted 3 pril 2011 vailable online 22 pril 2011 Communicated by Rudolf Lidl MSC: T06 Using a lemma proved by kbary, Ghioca, and Wang, we derive several theorems on permutation polynomials over finite fields. These theorems give not only a unified treatment of some earlier constructions of permutation polynomials, but also new specific permutation polynomials over F q. number of earlier theorems and constructions of permutation polynomials are generalized. The results presented in this paper demonstrate the power of this lemma when it is employed together with other techniques Elsevier Inc. ll rights reserved. Keywords: Permutation polynomials Finite fields Commutative diagrams 1. Introduction Let q be a prime power, F q be the finite field of order q, and F q [x] be the ring of polynomials in a single indeterminate x over F q. polynomial f F q [x] is called a permutation polynomial PP) of F q if it induces a one-to-one map from F q to itself. Permutation polynomials over finite fields have been an interesting subject of study for many years, and have applications in coding theory [12], cryptography [20,19], combinatorial design theory [10], and other areas of mathematics and engineering. Information about properties, constructions, and applications of permutation polynomials may be found in Lidl and Niederreiter [13,14], and Mullen [17]. * Corresponding author. Fax: address: cding@ust.hk C. Ding). 1 This research was supported by the NSF of China under Grant No /$ see front matter 2011 Elsevier Inc. ll rights reserved. doi: /j.ffa

2 P. Yuan, C. Ding / Finite Fields and Their pplications ) The trace function Trx) from F q n to F q is defined by Trx) = x + x q + x q2 + +x qn 1. number of classes of permutation polynomials related to the trace functions were constructed [6,5,4, 11,15,25]. Recently, kbary, Ghioca and Wang proved a lemma about permutations on finite sets [1,2], which contains two results of Zieve [23, Lemma 2.1] and [25, Proposition 3]) as special cases [16], and used this lemma to unify some earlier constructions and developed new constructions of permutation polynomials over finite fields. In this paper, we will employ this lemma to derive several theorems about permutation polynomials over finite fields. These theorems give not only a further unified treatment of some of the earlier constructions of permutation polynomials, but also new specific permutation polynomials. The main contributions of this paper are Theorems 3.1, 5.1, 6.1, 6.4, their corollaries and specific permutation polynomials described in the corollaries. 2. uxiliary results and the powerful lemma In this section, we present some auxiliary results that will be needed in the sequel. Throughout this paper p is a prime and q = p e for a positive integer e. polynomial of the form n 1 Lx) = a i x qi F q n[x] i=0 is called a q-polynomial over F q n, and is a permutation polynomial on F q n matrix a 0 a 1 a 2 a n 1 a q n 1 a q 0 a q 1 a q n 2 = a q2 n 2 a q2 n 1 a q2 0 a q2 n 3 a qn 1 1 a qn 1 2 a qn 1 3 a qn 1 0 if and only if the circulant 2.1) has nonzero determinant see [9, p. 362]). In most cases it is not convenient to use this result to find out permutation q-polynomials, as it may be hard to determine if the determinant of this matrix is nonzero [9]. Hence it would be interesting to develop other approaches to the construction of permutation q-polynomials. We shall use the following trivial fact in the sequel. Lemma 2.1. Let Lx) = n 1 i=0 a ix qi F q [x] be a q-polynomial and let Trx) be the trace function from F q n to F q. Then, for each α F q n,wehave L Trα) ) = Tr Lα) ) = Definition 2.2. See [14, p. 115].) The polynomials n 1 ) a i Trα). i=0 m m lx) = a i x i and Lx) = a i x qi i=0 i=0

3 562 P. Yuan, C. Ding / Finite Fields and Their pplications ) over F q n are called the q-associate of each other. More specifically, lx) is the conventional q-associate of Lx) and Lx) is the linearized q-associate of lx). The following lemma is also needed in the sequel. Lemma 2.3. See [14, p. 109].) Let L 1 x) and L 2 x) be two q-polynomials over F q,andletl 1 x) and l 2 x) be the q-associate polynomials over F q. Then the common roots of L 1 x) = 0 and L 2 x) = 0 are all the roots of the linearized q-associate of gcdl 1 x),l 2 x)). In particular, x = 0 is the only common root of L 1 x) = 0 and L 2 x) = 0 in any finite extension of F q if and only if gcdl 1 x),l 2 x)) = 1. The following lemma is taken from kbary, Ghioca, and Wang [1, Lemma 1.1], and will be employed in the sequel. For completeness and an easy understanding of later applications, we provide its proof here. Lemma 2.4. See [1, Lemma 1.1].) Let, Sand S befinitesetswith S = S, and let f :, h : S S, λ : S, and λ : S be maps such that λ f = h λ. Ifbothλ and λ are surjective, then the following statements are equivalent: i) fisbijectivea permutation of ); and ii) h is bijective from S to S and if f is injective on λ 1 s) for each s S. Proof. We have the following commutative diagram f λ λ S h S ssume first that f is bijective. Then f is injective on each λ 1 s). Furthermore, because λ and f are surjective and λ f = h λ, thenh : S S is surjective and thus bijective because S and S are finite sets of the same cardinality). Conversely, assume f a 1 ) = f a 2 ) for some a 1, a 2. Then hλa 1 )) = λ f a 1 )) = λ f a 2 )) = hλa 2 )). s h is a bijection, we obtain λa 1 ) = λa 2 ). Hence a 1, a 2 are in λ 1 s) for some s S. Since f is injective on each λ 1 s), we conclude that a 1 = a 2.So f is injective and in fact bijective because is a finite set). 3. The first theorem about permutation polynomials Our objective in this section is to present a general result on permutation polynomials which contains some earlier results as special cases. This gives a uniform treatment of some earlier constructions of permutation polynomials and also new permutation polynomials. The following theorem is derived from Lemma 2.4, and is a generalization of Theorems 1.5 and 6.1 in [1]. Theorem 3.1. Let q be a prime power, and let r 1 and n 1 be positive integers. Let Bx), L 1 x),...,l r x) F q [x] be q-polynomials, gx) F q n[x],h 1 x),...,h r x) F q [x] and δ 1,...,δ r F q n such that Bδ i ) F q and h i BF q n)) F q.then f x) = g Bx) ) + is a permutation polynomial of F q n if and only if r ) ) Li x) + δ i hi Bx)

4 P. Yuan, C. Ding / Finite Fields and Their pplications ) ) Bgx)) + r L ix) + Bδ i ))h i x) permutes BF q n); and 2) for any y BF q n), r L ix)h i y) permutes kerb). Moreover, 2) is equivalent to gcd r l ix)h i y), bx)) = 1 for any y F q,wherel i x) and bx) are the conventional q-associate of L i x) and Bx). Proof. Note that Bx) is a q-polynomial over F q. We have abx) = Bax) for all a F q and all x F q n, and Bx + y) = Bx) + By) for all x and y in F q n.hencebf q n) is a linear subspace of F q n over F q.sincebx) and L i x) are q-polynomials over F q,wehavel i Bx)) = BL i x)). By assumption, Bδ i ) F q for all i. It follows that the following polynomial hx) := B g)x) + r Li x) + Bδ i ) ) h i x) induces a mapping from BF q n) to itself. Since h i BF q n)) F q and B, L 1, L 2,...,L r are F q -linear over F q,wehaveb f = h B. Hencewe have the following diagram F q n f F q n B BF q n) h B BF q n) We would now apply Lemma 2.4 with = F q n, f = g B + r L i + δ i )h i B, S = S = BFq n), λ = λ = B and h as defined above. By Lemma 2.4, f x) is a permutation polynomial of F q n if and only if hx) induces a permutation of BF q n) and f x) is injective on each B 1 y) F q n for all y BF q n). For any given y BF q n) and α,β B 1 y), wehaveβ = α + x for some x ker B. Since f β) = f α + x) = g Bα) ) r ) ) r + Li α + x) + δ i hi Bα) = f α) + L i x)h i y), f x) is injective on each B 1 y) F q n for all y BF q n) if and only if r ) ker B ker L i h i y) ={0} for all y BF q n). The last conclusion of this theorem then follows from Lemma 2.3. In what follows in this section, we show that some earlier results on permutation polynomials are special cases of Theorem 3.1 and present new results on permutation polynomials. Corollary 3.2. See [1, Theorem 6.1].) Let q be a prime power, and let L 1, L 2, L 3 : F q n F q n be q-polynomials over F q.letwx) F q n[x] such that wl 3 F q n)) F q.then f x) = L 1 x) + L 2 x)w L 3 x) ) is a permutation polynomial of F q n if and only if the following two conditions hold:

5 564 P. Yuan, C. Ding / Finite Fields and Their pplications ) i) kerf y ) kerl 3 ) ={0}, for any y L 3 F q n),where F y x) := L 1 x) + L 2 x)wy). ii) hx) := L 1 x) + L 2 x)wx) permutes L 3 F q n). Proof. The conclusion of this corollary follows from Theorem 3.1 by setting r = 2, g = 0, h 1 = 1, h 2 = w, δ 1 = δ 2 = 0, B = L 3. The following is a consequence of Theorem 3.1. Corollary 3.3. Let r 1 and n 1 be positive integers. Let L 1 x),...,l r x) F q [x] be q-polynomials, gx) F q n[x],h 1 x),...,h r x) F q [x] and δ 1,...,δ r F q n.then F x) = g Trx) ) + is a permutation polynomial of F q n if and only if r ) ) Li x) + δ i hi Trx) 1) Trgx)) + r L ix) + Trδ i ))h i x) permutes F q ; and 2) for any y F q, r L ix)h i y) permutes kertr). Moreover, 2) is equivalent to gcd r l ix)h i y), n 1 i=0 xi ) = 1 for any y F q,wherel i x) is the conventional q-associate of L i x). Proof. Put Bx) = Trx) intheorem3.1.wehavethenbf q n) = TrF q n) = F q.hencebδ i ) F q.the conclusion of this corollary follows from Theorem 3.1. The following is a consequence of Corollary 3.3 and Lemma 2.3. Corollary 3.4. Let r 1 and n 1 be positive integers. Let L i x) = n 1 j=1 ai) x q j, i= 1,...,r, be q- j polynomials over F q,gx) F q n[x],h 1 x),...,h r x) F q [x] and δ 1,...,δ r F q n.then F x) = g Trx) ) + r ) ) Li x) + δ i hi Trx) is a permutation polynomial of F q n if and only if the following two conditions hold: 1) Trgx)) + r n 1 j=1 ai) x + Trδ j i ))h i x) is a permutation polynomial of F q. 2) For any y F q, the only common solution of the two equations r L ix)h i y) = 0 and Trx) = 0 in F q n is x = 0. Moreover, 2) is equivalent to gcd r l ix)h i y), n 1 i=0 xi ) = 1 for any y F q,wherel i x) is the conventional q-associate of L i x). Proof. By Lemma 2.3, gcd r l ix)h i y), n 1 i=0 xi ) = 1foranyy F q if and only if condition 2) is satisfied. The desired conclusion then follows from Corollary 3.3.

6 P. Yuan, C. Ding / Finite Fields and Their pplications ) Corollary 3.5. Let Lx) = n 1 i=0 a ix qi F q [x] be a q-polynomial, and let α F q n.then L α x) = α Trx) + Lx) is a permutation polynomial of F q n if and only if n 1 i=0 a i + Trα) 0 and gcdlx), n 1 i=0 xi ) = 1,wherel i x) is the conventional q-associate of L i x). Proof. In Corollary 3.4, put gx) = αx, r = 1, L 1 x) = Lx), δ 1 = 0, and h 1 x) = 1. We have then g Trx) ) + On the other hand, we have r ) ) Li x) + δ i hi Trx) = α Trx) + Lx) = Lα x). Jx) := Tr gx) ) + r Li x) + Trδ i ) ) h i x) = Trαx) + Lx). Clearly, Jx) maps F q to F q. Restricting Jx) on F q,wehave Jx) := Trαx) + Lx) = Trα)x + n 1 a i )x. i=0 The desired conclusion then follows from Corollary 3.4. The following follows from Corollary 3.5 directly. Corollary 3.6. Let Lx) = n 1 i=0 a ix qi F q [x] be a q-polynomial over F q.letα F q n such that Trα) = 0. Then L α x) = α Trx) + Lx) is a permutation polynomial of F q n if and only if gcdlx), x n 1) = 1. In particular, Lx) is a permutation polynomial of F q n if and only if gcdlx), x n 1) = 1,wherelx) is the conventional q-associate of Lx). In Theorem 3.1, putting L 1 x) = L 2 x) = =L r x) = Lx), we obtain the following. Corollary 3.7. Let q be a prime power, and let r 1 and n 1 be positive integers. Let Bx), Lx) F q [x] be q-polynomials, gx) F q n[x], h 1 x),...,h r x) F q [x] and δ 1,...,δ r F q n such that Bδ i ) F q and h i BF q n)) F q.then f x) = g Bx) ) + is a permutation polynomial of F q n if and only if r ) ) Lx) + δi hi Bx) 1) Bgx)) + r Lx) + Bδ i))h i x) permutes BF q n); and 2) gcd r h iy))lx), bx)) = 1 for any y BF q n), wherel i x) and bx) are the conventional q- associate of L i x) and Bx).

7 566 P. Yuan, C. Ding / Finite Fields and Their pplications ) The following result in [1] is a special case of Corollary 3.7. Corollary 3.8. See [1, Theorem 5.5].) Let q be a prime power, a F q,andletb F q n.letpx) be a q- polynomial over F q which permutes F q n,andlx) be a q-linear polynomial over F q.letgx) F q n[x] such that glf q n)) F q.then is a permutation polynomial over F q n if and only if f x) = apx) + Px) + b ) g Lx) ) i) a / glf q n)); and ii) hx) = apx) + Px) + Lb))gx) permutes LF q n). 4. Specific permutation polynomials from the first theorem In this section, we present a few classes of specific permutation polynomials that are consequences of Theorem 3.1. The following corollary is a generalization of Theorem 5 in [6]. Corollary 4.1. ssume that gcdn, k) = 1, gcdn, q) = 1 and δ F q n is an element with Trδ) = 0. Then the polynomial F X) = ax qk + x + δ) Trx) ) q k 1 is a permutation polynomial of F q n for all a F q \{0, 1}. Proof. We apply Corollary 3.4 with gx) = 0, r = 2, L 1 x) = ax qk, L 2 x) = x, δ 1 = 0, δ 2 = δ, h 1 x) = 1, and h 2 x) = x qk 1.Then F x) := g Trx) ) + Note that r ) ) Li x) + δ i hi Trx) = ax q k + x + δ) Trx) ) q k 1. Jx) := Tr gx) ) + Restricting Jx) on F q,wehave r Li x) + Trδ i ) ) h i x) = ax qk + x Trx) ) q k 1. because gcdn, q) = 1. Hence Jx) permutes F q. On the other hand, we have Jx) := a + 1)x r L i x)h i y) = ax qk + xy qk 1. If there is an element y F q and an element x F q n such that Trx) = 0 and ax qk + xy qk 1 = 0

8 P. Yuan, C. Ding / Finite Fields and Their pplications ) which implies x = 0orx = by for some 0 b F q due to gcdk,n) = 1. If the latter equality holds, we have Trx) = nby = 0, and thus y = 0 and x = 0. Therefore the conditions of Corollary 3.4 are satisfied and the desired conclusion follows from Corollary 3.4. Corollary 4.2. Let n be a prime and let q = 2.Defineforeachiwith0 i < n { 1 if i is a quadratic residue modulo n, a i = 0 otherwise. Let Lx) = n 1 i=0 a ix qi F q [x] and L α x) = α Trx) + Lx), where α F q n. Then L α x) is a permutation polynomial of F q n n 5 mod 8) and Trα) = 1. if n 3 mod 8) and Trα) = 0 or Proof. By definition, Trα) + n 1 i=0 a i = 1inbothcases.Letlx) be the conventional q-associate of Lx). It was proved in [8] that gcdx n 1,lx)) = 1ifn 3 mod 8) and gcdx n 1,lx)) = x 1if n 5 mod 8). The desired conclusions then follow from Corollary 3.5. Corollary 4.3. Let n = n 1 n 2,wheren 1 and n 2 are two distinct primes such that gcdn 1 1,n 2 1) = 2, and let q = 2.Defineforeachiwith0 i < n 0, i {0,n 2, 2n 2,...,n 1 1)n 2 }, a i = 1, i {n 1, 2n 1,...,n 2 1)n 1 }, 1 i n 1 ) i n 2 ))/2, otherwise, 4.1) where a n 1 ) denotes the Legendre symbol. Let Lx) = n 1 i=0 a ix qi F q [x] and L α x) = α Trx) + Lx), where α F q n with Trα) = 1. ThenL α x) is a permutation polynomial of F q n if n 1 1 mod 8) and n 2 3 mod 8) or n 1 5 mod 8) and n 2 7 mod 8). Proof. By definition, Trα) + n 1 i=0 a i = 1 in both cases. Let lx) be the conventional q-associate of Lx). It was proved in [7] that gcdx n 1,lx)) = x 1 in both cases. The desired conclusions then follow from Corollary The second theorem and its applications The following theorem is another application of Lemma 2.4, and is a multiplication version of Theorem 1.4 in [1]. Theorem 5.1. ssume that is a finite field and S is a subset of such that the map λ : S is surjective. Let g :, h : S S, and f : be maps such that the following diagram commutes:

9 568 P. Yuan, C. Ding / Finite Fields and Their pplications ) f x)gλx)) λ λ S h S Then the map px) = f x)gλx)) permutes if and only if the following conditions hold. i) hisabijectionfromλ) to λ). ii) gy) 0 for every y λ) with λ 1 y)>1. iii) f x) is injective on each λ 1 y) for all y λ). Proof. By Lemma 2.4 and the assumptions of this theorem, f x)gλx)) permutes if and only if h is a bijection from λ) to λ) and f x)gλx)) is injective on each λ 1 y) for all y λ). For given y λ) and α,β λ 1 y), we haveλα) = λβ) = y. Define px) = f x)gλx)). If pα) = pβ), then pα) = f α)gy) = f β)gy) = pβ). Therefore px) is injective on each λ 1 y) for all y λ) if and only if gy) 0foreveryy λ) with λ 1 y)>1 and f x) is injective on each λ 1 y) for all y λ). The following is a consequence of Theorem 5.1 and Corollary 11 in [15]. It can be also derived from Theorem 6.3 in [1]. This demonstrates that Theorem 5.1 is a generalization of Corollary 11 in [15]. Corollary 5.2. If hx) F q [x], h0) 0, then the polynomial px) = xhtrx)) permutes F q n if and only if ux) = xhx) permutes F q. Proof. Define = F q n and λx) = Trx) = x + x q + +x qn 1. Since the following diagram F q n xhtrx)) F q n Tr Tr F q xhx) F q commutes, the desired conclusion follows from Theorem 5.1. Corollary 5.2 is related to [3,18,21,24]. Note that the following diagram commutes: F q x r gx s ) F q x s x s x s F q ) x s F q ) x r gx) s

10 P. Yuan, C. Ding / Finite Fields and Their pplications ) Note also that α r = β r, α s = β s implies α = β if and only if gcdr, s, q 1) = 1. The following follows from Corollary 5.2. Corollary 5.3. See [1, Proposition 3.1].) Let r and s be positive integers. Then x r gx s ) is a permutation polynomial of F q if and only if gcdr, s, q 1) = 1 and x r gx) s permutes F q )s. Combining Corollaries 5.2 and 5.3, we have the following corollary, which is an improvement of Proposition 2.12 in [22]. Corollary 5.4. ssume q 1 mod d). Ifhx) F q [x],h0) 0, then the polynomial px) = xhtrx) q 1)/d ) permutes F q n if and only if xhx) q 1)/d permutes μ d, which is the set of all dth roots of unity in F q. The permutation polynomials in Corollary 5.4 are related to permutation polynomials from cyclotomy [23,25]. For any integer d 2, define h d x) = x d 1 + x d 2 + +x + 1 F q [x]. Corollary 5.5. Let 2 d < q 1 such that d q 1), u 1,r 1 and 0 k i d 1, s i Z, i= 1,...,r. Let b i F q,i= 1,...,r, and polynomials g i x) F q [x], i= 1,...,r, be divisible by h d x).thepolynomial gx) = x u r ) gi x q 1)/d + ) b i x k iq 1)/d s i Fq [x] is a permutation polynomial if and only if the following four conditions hold: 1) b 1 b r 0 in F q. 2) gcdu,q 1)/d) = 1. 3) gcdu + r k is i )q 1)/d, d) = 1. 4) r g i1)/b i + 1) s i is a dth power in F q. Proof. By convention, we use μ d to denote the set of all dth roots of unity in F q.itiseasytocheck that the following diagram commutes: F q gx) x s F q ) s h x s F q ) s where s = q 1)/d and hx) = x u Since h1) = r g i1) + b i ) s iq 1)/d and r gi x) + ) b i x k s i i q 1)/d.

11 570 P. Yuan, C. Ding / Finite Fields and Their pplications ) hx) = x u r ) r bi x k s i i q 1)/d = b i ) siq 1)/d x u+ r k i s i )q 1)/d 5.1) for any x μ d \{1}, we have condition 1). It follows from Corollary 5.3 that gcdu,q 1)/d, q 1) = 1, i.e., gcdu,q 1)/d) = 1. So we get condition 2). By 5.1), hx) permutes x s F q ) if and only if r ) gcd u + k i s i )q 1)/d, d = 1 and r g i1)/b i + 1) s i is a dth power in F q. These are conditions 3) and 4). We are done. Corollary 5.5 above is a generalization of Theorem 1 in [25] and Theorem 2 in [15]. 6. Two more theorems on permutation polynomials The following theorem is another application of Lemma 2.4, and is a variant of Theorem 1.4c) and Theorem 5.1c) in [1]. Theorem 6.1. ssume that is a finite field and S, S are finite subsets of with S) = S) such that the maps ψ : Sand ψ : S are surjective and ψ is additive, i.e., ψx + y) = ψx) + ψy), x, y. Let g : S, and f : be maps such that the following diagram commutes: f +g ψ ψ ψ S f S and ψgψx))) = 0 for every x. Then the map px) = f x) + gψx)) permutes if and only if f permutes. Proof. It follows from Lemma 2.4 and the assumptions of this theorem that f x) + gψx)) permutes if and only if f is a bijection from S to S and f x) + gψx)) is injective on each ψ 1 s) for all s S. On the other hand, by assumption we have ψgψx))) = 0foreveryx. Hence, ψ f x) + g ψx) )) = ψ f x) ) + ψ g ψx) )) = ψ f x) ) for all x. Therefore, the following diagram commutes: f ψ ψ S f S

12 P. Yuan, C. Ding / Finite Fields and Their pplications ) pplying Lemma 2.4 to this commutative diagram, we know that f x) permutes if and only if f is a bijection from S to S and f x) is injective on each ψ 1 s) for all s S. Let a 1 ψ 1 s) and a 2 ψ 1 s), where s S. Note that ψa 1 ) = ψa 2 ) = s. Wehavethen and f a 1 ) + g ψa 1 ) ) = f a 1 ) + gs) f a 2 ) + g ψa 2 ) ) = f a 2 ) + gs). It follows that f x) + gψx)) is injective on each ψ 1 s) for all s S if and only if f x) is injective on each ψ 1 s) for all s S. Summarizing the discussions above proves the desired conclusion. We will employ Theorem 6.1 to prove the following corollary. Corollary 6.2. Let n and k be positive integers such that gcdn, k) = d > 1, let s be any positive integer with sq k 1) 0 mod q n 1). Let L 1 x) = a 0 x + a 1 x qd + a 1 x q2d + +a n/d 1 x qn d, a i F q, be a q d -polynomial with L 1 1) = 0 and let L 2 x) F q [x] be a linearized polynomial and gx) F q n[x].then permutes F q n if and only if L 2 x) permutes F q n. f x) = g L 1 x) )) s + L2 x) Proof. Since gcdk,n) = d, sogcdq k 1, q n 1) = q d 1, and thus sq k 1) 0 mod q n 1) if and only if sq d 1) 0 mod q n 1). Therefore sq dr sq dr 1) s mod q n 1 ). It follows that for any x F q n, gl 1 x))) sqrd = gl 1 x))) s for any positive integer r. Hence L 1 gl 1 x))) s ) = L 1 1)gL 1 x))) s = 0 for every x F q n. Now applying Theorem 6.1 with ψ = ψ = L 1 x) and f = L 2 x), we obtain the desired conclusion. Note that Corollary 6.2 can also be derived from Theorem 2 and Theorem 1 in [11]. In Corollary 6.2 putting L 1 x) = x qk x, gx) = x + δ,δ F q n and L 2 x) = x, we obtain the following corollary, which is a slight generalization of Proposition 2 and Remark 1 in [22] as in the following corollary q could be any prime power. Corollary 6.3. Let n and k be positive integers such that gcdn, k) = d > 1, and let s be any positive integer with sq k 1) 0 mod q n 1).Then permutes F q n for any δ F q n. hx) = x qk x + δ ) s + x Theorem 6.4. Let q = p e for some positive integer e.

13 572 P. Yuan, C. Ding / Finite Fields and Their pplications ) a) If k 2 is an even integer or k is odd and q is even, then f a,b,k x) := ax q + bx + x q x) k,a, b F q 2 with a + b F q,permutesf q 2 if and only if b aq. b) If k and q are odd positive integers, then f a,k x) := ax q +a q x+x q x) k,a F and a+a q 0,permutes q 2 F q 2 if and only if gcdk, q 1) = 1. Proof. Note that fa,b,k x) ) q fa,b,k x) = b a q) x ) q b a q ) x + x x q) k x q x ) k. a) Since k is even or k is odd and q is even, we have fa,b,k x) ) q fa,b,k x) = b a q) x ) q b a q ) x. On the other hand, since a + b, a + a q F q,thenb a q F q, and so b a q ) x q x ) = b a q) x ) q b a q ) x. Hence in case a) we have the following commutative diagram F q 2 f a,b,k x) Fq 2 x q x x q x S b a q )x S where S ={α q α, α F q 2} and S ={b a q )s: s S}. Note that s S if and only if Trs) = 0. It is obvious that b a q )x is a bijection from S to S if and only if b q a 0. For any x, y F q 2 with x q x = y q y and ax q + bx = ay q + by, we have x y) q = x y, ax y) q = bx y). Therefore by Lemma 2.4, f a,b,k x) := ax q + bx + x q x) k, a, b F q 2 with a + b F q,permutesf q 2 if and only if b a q. b) Since k is odd, we have the following commutative diagram F q 2 f a,b,k x) Fq 2 x q x x q x S 2x k S where S ={α q α, α F q 2} and S ={ 2s k : s S}.

14 P. Yuan, C. Ding / Finite Fields and Their pplications ) For any x, y F q 2, a F q 2 with a + a q 0, x q x = y q y and ax q + a q x = ay q + a q y hold only when x = y. Therefore by Lemma 2.4, f a,k x) := ax q +a q x+x q x) k, a F q 2 with a +a q 0, permutes F q 2 if and only if 2x k is a bijection from S to S. Let ɛ F q 2 \ F q such that ɛ q 1 = 1sinceq is odd). Then each α F q 2 is uniquely written as u + vɛ for some u, v F q. We compute then easily that α q α = 2vɛ. Sincek is odd, so 2vɛ) k = 2dɛ) k implies v = d if and only if gcdk, q 1) = 1. Therefore f a,k x) := ax q +a q x + x q x) k, a F q 2 with a + a q 0, permutes F q 2 if and only if gcdk, q 1) = 1. This completes the proof. Theorem 6.4 above is a generalization of Theorem 5.12 in [1] in the following aspects: 1. In case a), the constants a and b in Theorem 6.4 belong to F q 2, while in Theorem 5.12 of [1] the two elements a and b are in the subfield F q. 2. In case b), the constant a in Theorem 6.4 belongs to F q 2, while in Theorem 5.12 of [1] the constant a is in the subfield F q. 3. In Theorem 6.4 above, the case that k is odd and q is even dealt with, while this case is not considered in Theorem 5.12 of [1]. 7. Concluding remarks Lemma 2.4 does not require the sets, S and S to have any algebraic structures. s demonstrated in [1] and this paper, Lemma 2.4 can be employed to construct many types of permutation polynomials over finite fields when the sets, S and S are finite fields and subsets of finite fields. This clearly shows the power and potential of Lemma 2.4. Of course, one has to figure out specific techniques of using Lemma 2.4 in order to construct specific permutation polynomials. ll the results presented in this paper were obtained by using Lemma 2.4 and specific techniques in finite fields. cknowledgments The authors would like to thank the reviewers for their detailed comments that improved the quality and presentation of this paper. References [1]. kbary, D. Ghioca, Q. Wang, On constructing permutations of finite fields, Finite Fields ppl ) [2]. kbary, Q. Wang, generalized Lucas sequence and permutation binomials, Proc. mer. Math. Soc ) [3]. kbary, Q. Wang, On polynomials of the form x r f x q 1 /l), Int. J. Math. Math. Sci. 2007), rticle ID [4] P. Charpin, G. Kyureghyan, When does F X) + γ TrHX)) permute F p n?, Finite Fields ppl. 15 5) 2009) [5] P. Charpin, G. Kyureghyan, On a class of permutation polynomials over F 2 n, in: SET 2008, in: Lecture Notes in Comput. Sci., vol. 5203, Springer-Verlag, 2008, pp [6]. Blokhuis, R.S. Coulter, M. Henderson, C.M. O Keefe, Permutations amongst the Dembowski Ostrom polynomials, in: D. Jungnickel, H. Niederreiter Eds.), Finite Fields and pplications: Proceedings of the Fifth International Conference on Finite Fields and pplications, Springer-Verlag, 2001, pp [7] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields ppl ) [8] C. Ding, T. Helleseth, W. Shan, On the linear complexity of Legendre sequences, IEEE Trans. Inform. Theory ) [9] C. Ding, Q. Xiang, J. Yuan, P. Yuan, Explicit classes of permutation polynomials over GF3 3m ), Sci. China Ser ) [10] C. Ding, J. Yuan, family of skew Hadamard difference sets, J. Combin. Theory Ser ) [11] G. Kyureghyan, Constructing permutations of finite fields via linear translators, J. Combin. Theory Ser ) [12] Y. Laigle-Chapuy, Permutation polynomials and applications to coding theory, Finite Fields ppl ) [13] R. Lidl, H. Niederreiter, Finite Fields, second ed., Encyclopedia Math. ppl., vol. 20, Cambridge University Press, Cambridge, 1997.

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