Further results on complete permutation monomials over finite fields
aa r X i v : . [ m a t h . N T ] A ug FURTHER RESULTS ON COMPLETE PERMUTATIONMONOMIALS OVER FINITE FIELDS
XIUTAO FENG, DONGDAI LIN, LIPING WANG, AND QIANG WANG
Abstract.
In this paper, we construct some new classes of complete permu-tation monomials with exponent d = q n − q − using AGW criterion (a specialcase). This proves two recent conjectures in [41] and extends some of theserecent results to more general n ’s. Let q = p k be the power of a prime number p , F q be a finite field with q elements,and F q [ x ] be the ring of polynomials over F q . We call f ( x ) ∈ F q [ x ] a permutationpolynomial (PP) of F q if f induces a permutation of F q . An exceptional polynomial of F q is a polynomial f ∈ F q [ x ] which is a permutation polynomial over F q m forinfinitely many m . It is well known that a permutation polynomial over F q of degreeat most q / is exceptional over F q . For more background material on permutationpolynomials we refer to Chap. 7 of [23]. For a detailed survey of open questionsand recent results see [16], [21], [22], [25], and [26].A complete permutation polynomial (CPP) is a polynomial f ( x ) satisfies thatboth f ( x ) and f ( x ) + x induce bijections of F q . CPPs have recently become astrong source of interest due to their connection to combinatorial objects such asorthogonal Latin squares and due to their applications in cryptography; in partic-ular, in the construction of bent functions [28, 30, 31]. See also [14, 32, 33, 38] andthe references therein for some recent work in the area.The most studied class of CPPs are monomials. If there exists a complete per-mutation monomial of degree d over F q , then d is called a CPP exponent over F q .Earlier work has been done recently in [7, 8, 9, 10, 24, 40]. Thesel recent papersconcentrated on classifying monomials f ( x ) = a − x d with special d that are CPPs.In particular, d = q n − q − + 1 implies that ( d, q n −
1) = 1 and thus x d is always a PPof F q n . Essentially, this problem is equivalent to classifying permutation binomialof the form x d + ax . The main focus of the paper is therefore to determine whencertain polynomials of the form f ( x ) := x qn − q − +1 + bx are permutation binomialsover F q n . We remark earlier references on permutation binomials can be found in[17, 26] and reference therein.One of useful criterions in studying PPs of finite fields is the following AGWcriterion. It first appeared in [3] and further developed in [42, 43], among others. Mathematics Subject Classification.
Key words and phrases. finite fields, permutation polynomials, complete permutation polyno-mials, monomials.The research of authors is partially supported by National Natural Science Foundation of China(No. 61379139), National Natural Science Foundation of China (No. 61572491), FoundationScience Center project of the National Natural Science Foundation of China (No. 11688101), andNSERC of Canada.
Lemma 1 (AGW criterion) . Let A , S and ¯ S be finite sets with S = S , and let f : A → A , ¯ f : S → ¯ S , λ : A → S , and ¯ λ : A → ¯ S be maps such that ¯ λ ◦ f = ¯ f ◦ λ . A P / / λ (cid:15) (cid:15) A ¯ λ (cid:15) (cid:15) S ¯ P / / ¯ S If both λ and ¯ λ are surjective, then the following statements are equivalent: (i) f is a bijection (a permutation of A ); and (ii) ¯ f is a bijection from S to ¯ S and f is injective on λ − ( s ) for each s ∈ S . We remark that P ( x ) in Lemma 1 can be viewed piecewisely. Namely, let S = { s , s , . . . , s ℓ − } , then we have P ( x ) = P ( x ) , if x ∈ C = λ − ( s );... ... P i ( x ) , if x ∈ C i = λ − ( s i );... ... P ℓ − ( x ) , if x ∈ C ℓ − = λ − ( s ℓ − ) , is a bijection if and only if each P i is injective on C i for 0 ≤ i ≤ ℓ − P is abijection.In particular, if we take A = F ∗ q = < γ > , ℓs = q − ζ = γ s , and λ = ¯ λ = x s ,then we obtain the following diagram. F ∗ q P / / x s (cid:15) (cid:15) F ∗ qx s (cid:15) (cid:15) S = { , ζ, . . . , ζ ℓ − } ¯ P / / S = { , ζ, . . . , ζ ℓ − } This special case of AGW criterion was obtained earlier in [27, 35]. Namely, thecyclotomic mapping polynomial P ( x ) = A x r , if x ∈ C = < γ ℓ > ≤ F ∗ q = < γ > ;... ... A i x r , if x ∈ C i = γ i C ;... ... A ℓ − x r , if x ∈ C ℓ − = γ ℓ − C , is a PP of F q if and only if ( r, s ) = 1 and ¯ P permutes S = { , ζ, . . . , ζ ℓ − } .Rewriting it in terms of polynomials, this criterion essentially is the followinguseful criterion appeared in [19, 35, 46] in different forms. Corollary 1 (Park-Lee 2001, Wang 2007, Zieve 2009) . Let q − ℓs for somepositive integers ℓ and s . Then P ( x ) = x r f ( x s ) is a PP of F q if and only if ( r, s ) = 1 and x r f ( x ) s permutes the set µ ℓ of all distinct ℓ -th roots of unity. Using Corollary 1, Akbary and Wang [6] first studied the permutation polyno-mials of the form x r f ( x s ) for arbitrary ℓ and those polynomials f ’s satisfying that f ( ζ ) s = ζ j for some j , where ζ ∈ µ ℓ . In particular, OMPLETE PERMUTATION MONOMIALS 3
Corollary 2.
Let q − ℓs . Assume that f ( ζ i ) s = 1 for any i = 0 , . . . , ℓ − and ζ ∈ µ ℓ is a primitive ℓ -th root of unity. Then P ( x ) = x r f ( x s ) is a PP of F q if andonly if ( r, q −
1) = 1 . More classes of those permutation polynomials with small ℓ and special f suchthat f ( x ) = x e + a s or f ( x ) = x k + x k − + · · · + x + 1 were studied earlier in[1, 4, 5, 6] and slightly extended in [45, 46]. PPs of large indices was studied in [37].For the intermediate indices, Zieve first considered special ℓ = q − ℓ = q + 1over finite field F q [47]; see also in [17]. We note that several recent papers dealwith PPs with this type of indices, see for example [20, 44].In the study of CPP monomials, Wu et al [40, 41] studied the CPP monomials a − x d over F p nk such that d = p nk − p k − + 1. For any a ∈ F p nk , let a i = a p ik , where0 ≤ i ≤ n −
1. Define h a ( x ) = x n − Y i =0 ( x + a i ) . Then Corollary 1 directly gives the following
Corollary 3.
Let d = p nk − p k − + 1 . Then x d + ax ∈ F p nk [ x ] is a PP of F p nk if andonly if h a ( x ) ∈ F p k [ x ] is a PP of F p k . In this case h a ( x ) = x ( x + a ) d − is a polynomial of lower degree over µ p k − or F p k . When n is small, essentially we need to study permutation polynomial of lowdegree over a subfield F p k . In [9, 10, 40, 41], PPs of the form f a ( x ) = x d + ax over F q n are thoroughly investigated for n = 2 , ,
4. For n = 6, sufficient conditions for f a ( x ) to be a PP of F q are provided in [40, 41] in the special cases of characteristic p ∈ { , , } , whereas in [7] all a ’s for which ax q − q − + 1 is a CPP over F q areexplicitly listed. The case p = n + 1 is dealt with in [41, 24] as well.The following two conjectures are made in [41]. Conjecture 1 (Conjecture 4.18 in [41]) . Let n + 1 be a prime such that n + 1 = p .Let ( n, k ) = 1 and ( n +1 , p −
1) = 1 , and d = p nk − p k − +1 . Then there exists a ∈ F ∗ p nk such that h a ( x ) are Dickson polynoials of degree n + 1 over F p k . Conjecture 2 (Conjecture 4.20 in [41]) . Let p be an odd prme. Let n + 1 = p and d = p nk − p k − + 1 . Then a − x d is a CPP over F p nk , where a ∈ F ∗ p nk such that a p k − = − . Using the classification results of exceptional polynomials, Bartoli et al [8] clas-sified complete permutation monomials of degree q n − q − + 1 over the finite field with q n elements in odd characteristic, for n + 1 a prime and ( n + 1) < q . As a corol-lary, Conjecture 1 was proven in odd characteristic. However, Conjecture 1 is stillopen in general. We note Conjecture 2 is proven recently in [24] for n + 1 prime.However, when n + 1 is large or not prime, the classification of CPP monomials isstill open. For example, when n + 1 is a power of primes such as 8 or 9, only a fewnew examples of CPPs are provided in [8]. In this paper, we construct several newclasses of CPPs using AGW criterion, which confirm both conjectures. In Section1, we use a factorization result of Dickson polynomial to construct explicitly a newclass of CPPs and prove Conjecture 1. In comparision to the proof in [8], our resultis more explicit although our result is not a classification result. However our proof FENG, LIN, WANG, AND WANG is elementary and it does not assume that ( n + 1) < p k . In Section 2, we derived afew new classes of CPPs for general n such that n | p − n | p k − n . We also demonstrated a usage of AGW criterion in proving a well known classof degree p exceptional polynomials.1. CPPs induced from Dickson polynomials
Let us consider Dickson polynomial of the first kind D n ( x, b ) of degree n over F q . It is well known that D n +1 ( x, b ) = D n +1 ( y + b/y, b ) = y n +1 + ( b/y ) n +1 whenwe let x = y + b/y for some y ∈ F q . Therefore w = u + bu is a root of D n +1 ( x, b )if and only if u n +1 + b n +1 u n +1 = 0. Equivalently, u is a solution to u n +1) = − b n +1 .If c = b and ζ is a primitive 4( n + 1)-th root of unity, then u is of the form cζ i +1 for 0 ≤ i ≤ n −
1. Therefore all the roots of D n +1 ( x, b ) are c ( ζ i +1 + ζ − (2 i +1) ) for0 ≤ i ≤ n . Moreover, the explicit factorization for Dickson polynomial of the firstkind was studied earlier in [12] and [11]. See also Theorem 9.6.12 in [13] as follows: Theorem 1. If q is odd and a ∈ F ∗ q then D n ( x, a ) is the product of irreduciblepolynomials over F q which occur in cliques corresponding to the divisors d of n forwhich n/d is odd. Let m d is the least positive integer satisfying q m d ≡ ± d ) .To each such d there corresponds φ (4 d ) / (2 N d ) irreducible factors of degree N d , eachof which has the form N d − Y i =0 ( x − p a q i ( ζ q i + ζ − q i )) where ζ is a primitive d -th root of unity and N d = m d / if √ a / ∈ F q , m d ≡ and q m d / ≡ d ± d ) , m d if √ a / ∈ F q and m d is odd ,m d otherwise . . We use this factorization result to construct a class of CPP as follows:
Theorem 2.
Let p be an odd prime, k be an odd positive integer, and n + 1 be anodd prime number such that ( n, k ) = 1 and ( n + 1 , p −
1) = 1 . Let d = p nk − p k − + 1 , b ∈ F ∗ p k , and c ∈ F p k such that b = c . Let a = c ( ζ + ζ − ) and ζ is a primitive n + 1) -th roots of unity in F p nk . Then(i) If p ≡ , or p ≡ and n/ is even, then a − x d is a CPPover F p nk .(ii) If p ≡ and n/ is odd, then a − x d is a CPP over F p nk for all c ∈ F p k \ F p k and b = c ∈ F p k .Proof. Let q = p k and m d be the smallest positive integer such that q m d ≡ ± n + 1)). By Fermat’s little theorem, we obtain p n ≡ n + 1) and p n/ ≡ − n + 1). Because ( n, k ) = 1, we must have k odd. If p ≡ q = p k ≡ m d such that q m d ≡ ± n + 1)) is m d = n . If p ≡ q ≡ − k is odd. Furthermore, if n/ m d such that q m d ≡ ± n + 1)) is m d = n/
2. If n/ m d such that q m d ≡ ± n + 1)) is m d = n . Inany case, a p nk = c p nk ( ζ p nk + ζ − p nk ) = c ( ζ + ζ − ) = a and thus a ∈ F p nk . OMPLETE PERMUTATION MONOMIALS 5 If p ≡ p ≡ n/ m d = n and thus byTheorem 1 we obtain, for any b ∈ F ∗ q , D n +1 ( x, b ) = x n − Y i =0 ( x − p b q i ( ζ q i + ζ − q i )) . If p ≡ n/ m d = n/
2. In this case, let b = c such that c ∈ F q \ F q . Then D n +1 ( x, b ) = x n − Y i =0 ( x − p b q i ( ζ q i + ζ − q i )) . In order to prove x d + ax is a permutation polynomial over F p nk , by Corollary 1, x d + ax is a PP of F p nk if and only if x ( x + a ) d − permutes F ∗ p k . We note that x ( x + a ) d − = x ( x + a ) p k + ··· + p ( n − k = x ( x + a )( x + a ) p k · · · ( x + a ) p ( n − k Therefore x ( x + a ) d − = D n +1 ( x, b ) under both assumptions. Because ( n, k ) = 1and ( n + 1 , p − n + 1 , p k −
1) = 1. Therefore D n +1 ( x, b ) permutes F ∗ p k . Hence we complete the proof. (cid:3) We remark Theorem 2 confirms Conjecture 1 by providing explicit choices of a .Our result is more explicit and it does not assume that ( n + 1) < p k which is thecase in [8]. 2. CPP exponent d = q n − q − with more general n In this section, we first use AGW criterion to give an elementary proof for theexceptionality of the class of degree p exceptional polynomials over F p m studied byFried, Guralnick and Saxl [15]. Use this class of exceptional polynomials, we derivea new class of complete permutation monomials a − x d over F q r with d = q p − − q − +1, q = p k , a ∈ F q n and a q − ∈ µ n \ { } where n | p −
1. This confirms and extendsConjecture 2. Similarly, we drive a class of complete permutation monomials over F q n with exponent d = q q − − q − + 1 and n | q −
1. This provides more examples ofCPPs with bigger values of n while previous studies mostly deal with small n ’s or n = p −
1. Finally a sufficient and necessary description for CPPs with d = p nk − p k − is provided. In particular, a simple class of such CPPs is given as a corollary. Lemma 2.
Let p be an odd prime, i ≥ , m, s be positive integers such that n | p − .If c ∈ F p m such that c ( p m − /n = 1 , then f ( x ) = x ( x n − c ) p − n is a permutationpolynomial over F p m . In particular, it is an exceptional polynomial over F p m for m ≥ .Proof. It is clear that f ( x ) = 0 if and only if x = 0. Let A = F ∗ p m , λ ( x ) = x n − c and ¯ λ ( x ) = x n , S = λ ( A ), ¯ S = ¯ λ ( A ), and ¯ f ( x ) = ( x + c ) x p − . To show f is a PPof F p m , by AGW criterion, we need to show that f is injective on λ − ( s ) for each s ∈ S and ¯ f is bijective.For each s ∈ S , we have λ − ( s ) is the set of all the distinct roots η , . . . , η n of x n = s + c . Hence f is injective on it because f ( η i ) = η i s p − n . FENG, LIN, WANG, AND WANG
To prove ¯ f is bijective from S to ¯ S , is is enough to show that ¯ f is surjective. Let b ∈ ¯ S such that ¯ f ( x ) = ( x + c ) x p − = x p + cx p − = b . Then x p + cx p − = b hasat least two distinct solutions if and only if y p − cb y − b has at least two distinctsolutions. Assume y , y are two distinct solutions in S . Then y − y is a root of y p − cb y = 0 and thus a root of y p − = cb , which is a contradiction because n | p − b ∈ ¯ S , and c is not an n -th power. Therefore there is at most one solution in S .Because | S | = | ¯ S | , we conclude ¯ f is a bijection. Moreover, f is an exceptionalpolynomial over F p m for m ≥ f is p . Hence we completethe proof. (cid:3) Theorem 3.
Let p be an odd prime, i ≥ , k, n be positive integers such that n | p − . Let d = p ( p − k − p k − + 1 and a ∈ F p rk such that a p k − ∈ µ n , where µ n isthe set of all n -th roots of unity in F p nk . If a p k − ∈ µ n \ { } , then a − x d is a CPPover F p nk .Proof. By Corollary 1 or Corollary 3, we need to prove h a ( x ) = x ( x + a ) d − is aPP of µ p k − (equivalently, a PP of F p k ). If a p k − ∈ µ n \ { } , then a p k = aζ forsome ζ = 1 ∈ µ n . Then a p ik = aζ i and h a ( x ) = x ( x + a ) d − = x ( x + a ) p k + ··· + p ( p − k = x ( x + a )( x + a ) p k · · · ( x + a ) p ( p − k = x ( x + a )( x + a p k ) · · · ( x + a p ( p − k )= x (cid:0) ( x + a )( x + aζ ) · · · ( x + aζ n − ) (cid:1) ( p − /n If a p k − = −
1, then h a ( x ) = x ( x − a ) ( p − / . We note that a p k − = − a ∈ F ∗ p k but a F p k . By Lemma 2, h a ( x ) is a PP of F p k and thus a − x d is aCPP over F p nk in this case.If a p k − ∈ µ n \ { , − } , then h a ( x ) = x ( x n + a n ) ( p − /n . We note that a p k − ∈ µ n \{ } implies that a n ∈ F ∗ p k but a F p k . If ( p k − /n is even, then ( − a n ) ( p k − /n = a p k − = 1. If ( p k − /n is odd, then ( − a n ) ( p k − /n = − a p k − = 1. Hence − a n isnot an n -th power in F p k . By Lemma 2, h a ( x ) is a PP of F p k and thus we completethe proof. (cid:3) We note that Conjecture 4.20 in [41] was confirmed by Theorem 3.2 [24]. It is aspecial case when n = p − a p k − = −
1. Our result deals with more general n such that n | p − a ’s such that a p k − ∈ µ n \ { , − } when n >
2. Next we extend the result for more general n ’s. Theorem 4.
Let p be an odd prime, k, n be positive integers such that n | p k − .Let d = p ( pk − k − p k − + 1 and ℓn = p k − . Let µ n be the set of all n -th roots of unityin F p nk . If a p k − ∈ µ n \ { } , then a − x d is a CPP over F p nk . OMPLETE PERMUTATION MONOMIALS 7
Proof.
Let a p k = aζ for some ζ = 1 ∈ µ n . Then a p ik = aζ i and h a ( x ) = x ( x + a ) d − = x ( x + a ) p k + ··· + p ( pk − k = x ( x + a )( x + a ) p k · · · ( x + a ) p ( pk − k = x ( x + a )( x + a p k ) · · · ( x + a p ( pk − k )= x (cid:0) ( x + a )( x + aζ ) · · · ( x + aζ n − ) (cid:1) ( p k − /n If a p k − = −
1, then h a ( x ) = x ( x − a ) ( p k − / . We note that a p k − = − a ∈ F ∗ p k but a F p k . For any p k − -th root of unity ξ , we have( ξ − a ) p k − = 1. By Corollary 2, h a ( x ) is a PP of F p k and thus a − x d is a CPPover F p nk . Similarly, If a p k − ∈ µ n \ { , − } , then h a ( x ) = x ( x n + a n ) ( p k − /n . Wenote that a p k − ∈ µ n \{ } implies that a n ∈ F ∗ p k but a F p k . Therefore x n + a n = 0has no solution in F p k . Hence ( ξ + a n ) p k − − ξ ∈ µ ℓ . By Corollary 2, h a ( x )is a PP of F p k and thus the result holds. (cid:3) The following description of CPP monomials is obvious for more general n | p k − Theorem 5.
Let p be an odd prime, n ≥ , k be positive integers such that n | p k − .Let d = p nk − p k − + 1 . Let a p k − be a primitive n -th root of unity in F p nk . Then a − x d is a CPP over F p nk if and only if [( a n + c i )( a n + c j ) − ] r = c j − i f or all ≤ i < j < ( p k − /n where c is a fixed primitive ( p k − n ) -th root of unity in F p k .Proof. Similarly as above, h a ( x ) = x ( x n + a n ). It is well known that h a ( x ) = x ( x n + a n ) is a PP of F p k if and only if ( − a n ) ( p k − /n = 1 and[( a n + c i )( a n + c j ) − ] n = c j − i f or all ≤ i < j < ( p k − /n where c is a fixed primitive ( p k − n )-th root of unity in F p k (see for example, Excercise7.11 in [23]) . Because a p k − is a primitive n -th root of unity in F p nk and n ≥ − a ) ( p k − /n = 1 always holds. (cid:3) Finally we obtain the following class of CPP monomials.
Corollary 4.
Let p be an odd prime, n ≥ , k be a positive integer such that nℓ = p k − . Let d = p nk − p k − + 1 . Let a p k − be a primitive n -th root of unity in F p nk .If there exist an integer λ such that ( z + a n /z ) n = z λ for every ℓ -th root of unity z , then a − x d is a CPP over F p nk if and only if (2 + n + λ, ℓ ) ≤ .Proof. As in Theorem 5, we need to prove h a ( x ) = x ( x n + a n ) is a PP of F p k . We useCorollary 1 again, h a ( x ) = x ( x n + a n ) is a PP of F p k if and only if g ( x ) = x ( x + a n ) n permute µ ℓ , the set of all ℓ -th roots of unity. Obviously, ( − a n ) ( p k − /n = 1. Then g ( x ) = x ( x + a n ) n permute µ ℓ if and only if g ( x ) = x ( x + a n ) n is injective on µ ℓ /µ . For any z ∈ µ ℓ , we must have g ( z ) = z ( z + a n ) n = z n ( z + a n /z ) n = FENG, LIN, WANG, AND WANG z n + λ . Therefore g ( x ) = x ( x + a n ) n is injective on µ ℓ /µ if and only if(2 + n + λ, ℓ ) ≤ (cid:3) References [1] A. Akbary, S. Alaric, and Q. Wang, On some classes of permutation polynomials,
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Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Sys-tems Science, Chinese Academy of Sciences, Beijing 100190, China
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E-mail address : [email protected] State Key Laboratory of Information Security, Institute of Information Engineer-ing, Chinese Academy of Sciences, Beijing 100093, China
E-mail address : [email protected] School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K S B , CANADA E-mail address ::