ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE

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1 ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG Abstract. We count permutation polynomials of F q which are sums of m + 2 monomials of prescribed degrees. This allows us to prove certain results about existence of permutation polynomials of prescribed shape.. Introduction Let p be prime and let q be a nontrivial power of p. A polynomial is a permutation polynomial of F q if it induces a bijective map from F q onto itself. The study of permutation polynomials of a finite field goes back to 9-th century when Hermite and later Dickson pioneered this area of research. In recent years, interests in permutation polynomials have significantly increased because of their potential applications in cryptography, coding theory, and combinatorics see for example [2], [8], [2]. For more background material on permutation polynomials we refer to Chapter 7 of [0]. In [9], Lidl and Mullen proposed nine open problems and conjectures involving permutation polynomials of finite fields. This is one of the open problems. Problem. Lidl-Mullen. Let N d q denote the number of permutation polynomials of F q which have degree d. We have the trivial boundary conditions: N q = qq, N d q = 0 if d is a divisor of q larger than, and N d q = q! where the sum is over all d < q such that d is either or it is not a divisor of q. Find N d q. Note that we may assume each polynomial defined over F q has degree at most q because x q = x for each x F q. We review several recent results regarding this problem. In [3], Das proved that N p 2 p ϕp/pp! as p, where ϕ is the Euler function. More precisely he proves that N p 2p ϕp p p! p p+ p 2 + p 2. p This result has been improved and generalized by Konyagin and Pappalardi [5] who proved that N q 2q ϕq q q! 2e π q q 2. Furthermore, Konyagin and Pappalardi [6] count the permutation polynomials which have no monomials of prescribed degrees. More precisely, they prove the following result Mathematics Subject Classification. T06, 05A6. Key words and phrases. Permutation polynomials, Finite fields. Research of the authors was partially supported by NSERC.

2 2 AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG Theorem.2 Konyagin-Pappalardi. Fix j integers k,..., k j with the property that 0 < k < < k j < q and define Nk,..., k j ; q as the number of permutation polynomials h of F q of degree less than q such that the coefficient of x k i in h equals 0, for i =,..., j. Then Nk,..., k j ; q q! q q j < + q k q q/2. e Note that N q 2 q = q! Nq 2; q. Theorem.2 leaves open the question that whether there are permutation polynomials with a prescribed set of nonzero monomials, as it only counts the permutation polynomials whose nonzero monomials are a subset of a given set of monomials. Moreover, Theorem.2 is vacuous when k is quite small comparing to q, as in that case the right hand side of the above inequality is larger than q!. In this paper we address both of the above issues. On one hand, we are able to count the permutation polynomials which have a prescribed set of nonzero monomials see Theorem.3. More precisely, we provide a partition of the set of permutation polynomials of degree d and obtain upper and lower bounds for the number of permutation polynomials in each class. On the other hand, as a consequence of our main result Theorem.3, we can prove the existence of many permutation polynomials which have a prescribed shape see Corollary.5, or a prescribed degree see Theorem.7. In order to state our results we need the following terminology. For any nonconstant monic polynomial gx F q [x] of degree q with g0 = 0, let r be the vanishing order of gx at zero and let f x := gx/x r. Then let l be the least divisor of q with the property that there exists a polynomial fx of degree l degf /q such that f x = fx q /l. So gx can be written uniquely as x r fx q /l. We cal the index of gx. More generally any nonconstant polynomial hx can be written as hx = agx + b where a 0 and gx is monic with g0 = 0. We define the index of hx as the index of gx. So any nonconstant polynomial hx F q [x] of degree q and of index l can be written uniquely as ax r fx q /l + b. Clearly, hx is a permutation polynomial of F q, if and only if gx = x r fx q /l is a permutation polynomial of F q. If l = then fx = and so gx = x r. In this case gx is a permutation polynomial of F q if and only if r, q =. So from now on we assume that l >. We set up the notation for our main result. Let q be a prime power, and l 2 be a divisor of q. Let m, r be positive integers, and ē = e,..., e m be an m-tuple of integers that satisfy the following conditions:. 0 < e < e 2 < e m l and e,..., e m, l = and r + e m s q, where s := q /l. For a tuple a := a,..., a m F q m, we let g a r,ex := x r x ems + a x e m s + + a m x e s + a m. Note that if r and e satisfy. then gr,e a x has index l. We observe that if gr,e a x is a permutation polynomial of F q then.2 r, s =,

3 ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE 3 and so, r + e m s < q. This is true, since otherwise r, s = c >. Let ω be a c-th root of unity in F q. Then gr,e a = ga r,e ω, and so ga r,e x is not a permutation polynomial. With the above notation define Nr,e m l, q as the number of all tuples a F m q such that gr,e a x is a permutation polynomial of F q. In other words Nr,e m l, q is the number of all monic permutation m + -nomials gr,e a x = xr fx q /l of F q with vanishing order at zero equal to r, set of exponents ē for fx, and index l. In this paper we will find an asymptotic formula for Nr,e m l, q. Our main result is the following. Theorem.3. We have l, q qm + q m /2 <. l! N m r,e In fact in 2.6 and 2.7 we establish more precise upper and lower bounds for the quotient in Theorem.3. Our theorem together with 2.6 and 2.7 improve and generalize Theorem 4.5 of [7] which treats only the case that m = and e =. We also note that one may generalize the methods introduced in [] to obtain similar bounds for Nr,e m l, q as in our Theorem.3. Next we employ Theorem.3 to study the existence of permutation polynomials of certain shapes. There are no permutation polynomials of F q of degree d > such that d q. Moreover Carlitz s conjecture now a theorem due to Fried, Guralnick, and Saxl [4] states that, for any positive even degree n, there is no permutation polynomial of degree n of F q if q is sufficiently large compared to n. On the other hand one can prove the existence of permutation polynomials of varying degrees, as it is evident from the following result. Theorem.4 Carlitz-Wells. i Let l >. Then for q sufficiently large such that l q, there exists a F q such that the polynomial xx q /l + a is a permutation polynomial of F q. ii Let l >, r, q =, and k be a positive integer. Then for q sufficiently large such that l q, there exists a F q such that the polynomial x r x q /l + a k is a permutation polynomial of F q. Note that in Theorem.4, part i is a special case of part ii for r = and k =. See [, Theorem 2 and Theorem 3] for a proof of the above result. Recently several authors found quantitative versions of the Carlitz-Wells theorem in binomial case. In [7], Laigle-Chapuy proves the first assertion of Theorem.4 assuming q > l 2l+2 + l In [], Masuda and Zieve obtain a stronger result for more general binomials of the form x r x e q /l + a. More precisely they show the truth of part i of Theorem.4 for q > l 2l+2. Here by employing Theorem.3 we obtain the following quantitative generalization of the Carlitz-Wells theorem, which surpasses all the above results. Corollary.5. For any q, r, ē, m, l that satisfy.,.2, and q > l 2l+2, there exists an ā F q m such that the m + -nomial gār,e x is a permutation polynomial of F q. Remark.6. For q 7 we have l 2l+2 < q as long as l < log q 2 log log q. As an immediate corollary of Theorem.3 we can obtain the existence of permutation m+- nomials which have coefficients equal to 0 for their x k terms, where 2 k s simply take r =

4 4 AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG in a permutation polynomial of the form x r fx s as in Corollary.5. This observation addresses one of the questions left open by Konyagin and Pappalardi see the discussion after Theorem.2. Next note that for t q 2 the number of permutation polynomials of degree at least q t is q! Nq t, q t,..., q 2; q. In [6, Corollary 2] Konyagin and Pappalardi proved that Nq t, q t,..., q 2; q q! q t holds for q and t q. This result will guarantee the existence of permutation polynomials of degree at least q t for t q as long as q is sufficiently large. Our next theorem gives generalization and refinement of this existence result for certain values of t. Theorem.7. Let m. Let q be a prime power such that q has a divisor l with m < l and l 2l+2 < q. Then for every t < l m l q coprime with q /l there exists an m + -nomial gār,e x of degree q t which is a permutation polynomial of F q. Note that Theorem.7 establishes the existence of permutation polynomials with exact degree q t. Our arguments in the proof of our main result Theorem.3 are of two flavors. Firstly, we use combinatorial arguments specific for permutation polynomials, which were inspired from Laigle- Chapuy s work [7]. Secondly, we use an upper bound on sums of multiplicative characters, which originally is a consequence of Weil s proof of the Riemann hypothesis for curves over finite fields. We further would like to point out that the proof of Theorem.3 for m > differs significantly from the case m =. In fact the direct application of the method of [7] leads to difficulties in the case m >. In this paper we deal with the latter case by splitting all tuples in F q m in two categories: good and bad tuples see Section 2 for the definition, and then applying the Weil bound only to the character sums associated to the good tuples. In the next section we prove our main result Theorem.3, and in Section 3 we prove Theorem Proof of the Main Theorem In the proof of Theorem.3 we will use the following classical inequality for character sums which is proved by Weil in [4] using methods of algebraic geometry see [0, Theorem 5.4] for an elementary proof; see also [0, Chapter 5] for a discussion of upper bounds for character sums. Theorem 2. Weil. Let Ψ be a multiplicative character of F q of order l > and let fx F q [x] be a monic polynomial of positive degree that is not an l-th power of a polynomial. Let d be the number of distinct roots of fx in its splitting field over F q. Then for every t F q we have Ψtfa a F q d q. We continue with the notation from Section ; so, l 2 is a divisor of q, and s = q /l. We denote by S l the set of all permutations of {,..., l}, and by µ l the set of al-th roots of unity in F q.

5 ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE 5 We let α be a generator of the cyclic group F q. Let ψ be a multiplicative character of order l of µ l. More precisely, let ω be a primitive l-th root of unity in C. Define ψα s = ω and extend it with ψ0 = 0. Finally, we let a multiplicative character Ψ of F q of order l be defined by Ψα = ψα s, and extended so that Ψ0 = 0. For any permutation σ S l, and any β,..., β l µ l, we define l l β,..., β l = ψβ i ψα s σi j. It is easy to see that i= 2. {β,..., β l } = µ l there exists σ such that β,..., β l =. j=0 Furthermore, there exists a unique permutation σ satisfying 2.; it is given by solving the equations 2.2 α s σi = β i, for each i {,..., l}. Indeed, it is known that for every l-th root of unity ζ we have l ζ k = 0, k=0 unless ζ =, in which case l k=0 ζk = l. So, if {β,..., β l } = µ l, then there exists a unique permutation σ β S l which satisfies 2.2 and then β β,..., β l =, as desired. Now, if {β,..., β l } = µ l, then for each permutation σ S l there exists some i =,..., l depending on σ such that 2.2 does not hold. Therefore, l ψβ i ψα s σi k = 0, k=0 and so, β,..., β l = 0. We summarize below our findings about. Lemma 2.2. Let β,..., β l µ l. Then β,..., β l = σ S l { if {β,..., β l } = µ l 0 otherwise We extend the definition of to β i µ l {0}. If there are exactly k numbers β i for some k {,..., l}, which are equal to 0, while the other β j s are in µ l, then for every σ S l we have 2.3 either β,..., β l = 0, or β,..., β l = k. In 2.3 we used the convention that 0 0 =. We will use the following result in our proof of Theorem.3. Lemma 2.3. If β i µ l {0} for each i l, and at least one β i is zero, then 0 σ S l β,..., β l l..

6 6 AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG Proof. If the nonzero β i s are not all distinct, then β,..., β l = 0 for every σ S l and the lemma is proved in this case. Hence we only need to consider that all nonzero β i s are distinct and k of the β i s are equal to 0. Now, assume without loss of generality, that β = = β k = 0 for some k {,..., l}, while β k+,..., β l are distinct elements of µ l. Then there are precisely k! permutations σ S l such that β,..., β l 0. Indeed, the value of σi for each i {k +,..., l} is determined by 2.2, while the values σi for i {,..., k} are arbitrary in the set {,..., l} \ {σk +,..., σl}. Thus, using 2.3, we obtain β,..., β l = k! l k l, σ S l as desired. We are ready to prove Theorem.3. Proof of Theorem.3. Because e and r are fixed satisfying. and.2, for the sake of simplifying the notation, we drop the indices r and e and denote gr,e a x by g a x = x r x ems + a x e m s + + a m x e s + a m. According to [3, Theorem.3], the polynomial g a permutes F q if and only if the following two conditions are satisfied: i α iems + a α ie m s + + a m α ie s + a m 0, for each i =,..., l; ii g a α i s g a α j s, for i < j l. Using conditions i and ii, and Lemma 2.2, we obtain 2.4 Nr,el, m q = g a α s,..., g a α l s. We note that there are at least and at most a F q m σ S l a satisfies i m m j j j=0 m m l j j j=0 q m j, q m j tuples a,..., a m F q m satisfying at least one of the equations α emis + a α e m is + + a m α e is + a m = 0, for some i l by using the inclusion-exclusion principle. An easy computation shows that m m q j m j = q m m. j q j=0

7 ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE 7 Using Lemma 2.3 in the formula 2.4 for each tuple a which fails condition i, we obtain g a α s,..., g a α l s q m m 2.5 q a F q m σ S l N m r,el, q a F q m σ S l g a α s,..., g a α l s. On the other hand, there are q m q m tuples a,..., a m F q m in which at least one a i = 0. Thus, using Lemma 2.2 in 2.5 for each tuple a which has at least one entry equal to 0, we obtain 2.6 a F m q = a F m q σ S l σ S l Nr,el, m q a F m q σ S l Now we consider two cases. Case : m > g a α s,..., g a α l s q m m q m q m q g a α s,..., g a α l s qm+ q m+ m g a α s,..., g a α l s. Let β := α s be a fixed generator of µ l. We call a m -tuple a,..., a m F q m good if there is no i < i 2 l such that 2.7 β i e m + a β i e m + + a m β i e = β i 2e m + a β i 2e m + + a m β i 2e. We call a m -tuple bad, if it is not good. Observe that the number of bad tuples is at least q m 2 and at most l 2 q m 2. Indeed, this follows from the fact that for each pair i, i 2 as above, 2.8 β i e m, β i e m,..., β i e β i 2 e m, β i 2e m,..., β i 2e, which is true since e,..., e m, l =. From 2.8 it follows that each equation 2.7 has at most q m 2 solutions in F q m. There could be no solutions if β i e i = β i 2e i for i m, while β i em β i 2e m. However, if i 2 i =, then 2.7 has precisely q m 2 solutions. An additional application of Lemma 2.2 for each bad tuple a,..., a m in 2.6 yields that g a α s,..., g a α l s qm+ q m+ m 2.9 q σ S l a,...,a m is good Nr,el, m q l q m + 2 σ S l a,...,a m is good g a α s,..., g a α l s. q

8 8 AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG For a fixed permutation σ S l, and a fixed good tuple a,..., a m, and a fixed set of numbers k,..., k l {0,..., l } not all equal to 0, and for every a m F q, we let M a,σ,k := l i= ψg a α i s ψα s σi k i, where a = a,..., a m and k = k,..., k l. We consider the sum M a,σ,k, where the tuple a,..., a m is fixed in the above summation. Using the multiplicitivity of ψ, the sum M a,σ,k equals 2.0 ψ β l i= rik i σik i l i= β e mi + a β em i + + a m β ei ki s + a m, which can be written as a character sum ψ t s l i= β e mi + a β em i + + a m β ei ki s + a m, where t := α l i= rik i σik i F q. Furthermore, using the previously defined character Ψ of F q of order l, the sum in 2.0 can be written as Ψ t l β e mi + a β em i + + a m β ei ki + a m. i= Because a,..., a m is a good tuple, and because k i < l for each i, we obtain that the monic polynomial R a,...,a m x := k l β e mi + a β em i + + a m β ei + x k i i= is not an l-th power of another polynomial. Let Ik := #{i : k i 0}. Using that a,..., a m is a good tuple, we conclude that R a,...,a m has Ik distinct roots. Because R a,...,a m is not an l-th power of another polynomial, we apply Theorem 2. and obtain k k that 2. Ψ t l β e mi + a β em i + + a m β ei ki + a m Ik q /2. i=

9 For each σ S l, we have ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE = a,...,a m is good a,...,a m is good + g a α s,..., g a α l s a,...,a m is good Ik M a,σ,k g a α s,..., g a α l s. Using the bounds for the number of bad tuples, we obtain 2.3 q m l q m 2 a,...,a m is good q m q m. On the other hand, 2.4 a,...,a m is good Ik M a,σ,k l M a,σ,k a,...,a m is good i= Ik=i g a α s,..., g a α l s g a α s,..., g a α l s, which by 2. is a,...,a m is good l i q /2 i= Ik=i q m q m 2 l l l i i q /2 i i= l l q q m 3/2 l i i i i= = + l 2 q q m 3/2,

10 0 AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG where in the above we used the fact that there are l i l i tuples k {0,..., l } l such that Ik = i. Using 2.3 and 2.4 in 2.2 for each permutation σ S l, we obtain l 2.5 q m q m + l 2 q q m 3/2 2 g a α s,..., g a α l s Applying 2.5 in 2.9 yields 2.6 Case 2: m = a,...,a m is good q m q m + + l 2 q q m 3/2. l! qm l! + l 2 q q m 3/2 qm+ q m+ m q Nr,el, m q l! qm + l! + l 2 q q m 3/2 + l 2 l! q m l l! 2 q m. The computation in this case is similar to m >, the only difference is that the bad tuples will not appear in this case. The final result is the following. l! 2.7 q l! + l 2 q /2 2 Nr,e l, q l! q + l! + l 2 q /2. Now let Cr,ēl, m l! q := N r,ēl, m q q m + q m /2. Observe that Nr,ēl, m 2 is not well defined, Nr,ē2, m 3 = 0, and Nr,ē3, m 4 = 0. So Cr,ēl, m q < if q = 3 or 4. So from now on we assume that q > 4. From 2.6 and 2.7 we have that for m >, 2l + l l q q l 2 q /2 q m+ q m+ m l l! C m r,ēl, q and for m =, 2l + l l q q + For m =, it is clear that 2l + l l l 2l! l l q /2, q m+/2 2 l l! q /2 C r,e l, q 2l + l l. C m r,e l, q < note that q > 4.

11 ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE Now we assume m >. For the upper bound of Cr,e m l, q we obtain 2l + l l q l + q 2l! < l q + q 2l q /2 <, l l q /2 as desired note that q > 4. For the lower bound for C m r,e l, q, using that l > e m m 2, we first obtain that = < Similarly, using that l 3, we get l l! qm+ q m+ m q m+/2 l l! qm + q m q q m m q m+/2 l l! m + q /2 l l l! q /2 2l q /2. l 2 q /2 < 2l q /2. Hence 2l + l l q q l 2 q /2 l l! > + l q q 2l q /2 2l q /2 = + q l q q /2 >, as desired since q > Existence of Permutation Polynomials Proof of Corollary.5. In Theorem.3 if N m r,e l, q = 0 then q < l2l+2. q m+ q m+ m q m+/2 Using Theorem.3 we can easily prove Theorem.7. Proof of Theorem.7. We look for a permutation polynomial of the form gx = x r fx s of degree q t where fx = x em + a x e m + + a m x e + a m,

12 2 AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG and s = q /l. This means that the degree e m of f satisfies the equation r + e m s = q t. Note that s > because l 2l+2 < q. Moreover since t < l ms and t is coprime with s, we can write t as t = u s + v, where 0 u < l m, and v {,..., s } is coprime with s. If m = we let r := l u s v and e m :=. Note that r > 0 since r = l s t and t < l s. It is clear that these choices for r and e m satisfy.,.2 and r +e m s = q t. Now since l 2l+2 < q, Corollary.5 implies the result in the case m =. If m 2 we let r := s v and e m := l u and choose m -tuple e,..., e m such that 0 < e < e 2 < e m < e m and 3. e,..., e m, e m, l =. Note that e m m and condition 3. is satisfied by various choices for e,..., e m such as e m = e m, or e =. It is easy to check that these values for r and ē = e,..., e m satisfy.,.2 and r + e m s = q t. Since l 2l+2 < q, Corollary.5 establishes the result for m 2. The following result is an immediate consequence of Theorem.7 for l = m + and t =. Corollary 3.. Let m be an integer, and let q be a prime power such that m + q. Then for all n 2m + 4, there exists a permutation m + -nomial of F q n of degree q n 2. References [] L. Carlitz and C. Wells, The number of solutions of a special system of equations in a finite field, Acta Arithmetica XII 966, [2] W. Chu and S. W. Golomb, Circular Tuscan-k arrays from permutation binomials, J. Combin. Theory Ser. A , no., [3] P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl , [4] M. D. Fried, R. Guralnick, J. Saxl, Schur covers and Carlitz s conjecture, Israel J. Math , no. 3, [5] S. Konyagin and F. Pappalardi, Enumerating permutation polynomials over finite fields by degree, Finite Fields Appl , no. 4, [6] S. Konyagin and F. Pappalardi, Enumerating permutation polynomials over finite fields by degree. II, Finite Fields Appl , no., [7] Y. Laigle-Chapuy, Permutation polynomials and applications to coding theory, Finite Fields Appl , no., [8] R. Lidl and W. B. Müller, Permutation polynomials in RSA-cryptosystems, Advances in Cryptology, Plenum, New York, 984, [9] R. Lidl and G. L. Mullen, When does a polynomial over a finite field permute the elements of the field?, Amer. Math. Monthly , [0] R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 997. [] A. Masuda and M. E. Zieve, Permutation binomials over finite fields, Trans. Amer. Math. Soc., to appear. [2] J. Sun and O. Y. Takeshita, Interleavers for Turbo codes using permutation polynomials over integer rings, IEEE Trans. Inform. Theory , no., 0-9.

13 ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE 3 [3] D. Wan and R. Lidl, Permutation polynomials of the form x r fx q /d and their group structure, Monatsh. Math. 2 99, [4] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U. S. A , Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB TK 3M4, Canada address: amir.akbary@uleth.ca Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB TK 3M4, Canada address: dragos.ghioca@uleth.ca School of Mathematics and Statistics, Carleton University, Ottawa, ON KS 5B6, Canada address: wang@math.carleton.ca