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Enumerating Permutation Polynomials over Finite Fields by Degree
Finite Fields and Their Applications 8, 548–553 (2002) doi:10.1006/ffta.2002.0363
Enumerating Permutation Polynomials over Finite Fields by Degree Sergei Konyagin1 Department of Mechanics and Mathematics, Moscow State University, Vorobjovy Gory, 119899 Moscow, Russia E-mail: [email protected],[email protected]
and Francesco Pappalardi Dipartimento Di Matematica, Universita" degli Studi Roma Tre, Largo S. L. Murialdo, 1, 1-00146 Rome, Italy E-mail: [email protected] Communicated by Wan Daging Received June 27, 2001; published online June 14, 2002
We prove an asymptotic formula for the number of permutations for which the associated permutation polynomial has degree smaller than q 2. # 2002 Elsevier Science (USA)
Let Fq be a finite field with q ¼ pf > 2 elements and let s 2 SðFq Þ be a permutation of the elements of Fq . The permutation polynomial fs of s is fs ðxÞ ¼
X
sðcÞð1 ðx cÞq1 Þ 2 Fq ½x:
c2Fq
fs has the property that fs ðaÞ ¼ sðaÞ for every a 2 Fq and this explains its name. For an account of the basic properties of permutation polynomials we refer to the book of Lidl and Niederreiter [5]. 1
To whom correspondence should be addressed. 548
1071-5797/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
ENUMERATING PERMUTATION POLYNOMIALS
549
From the definition, it follows that for every s @ðfs Þ q 2: A variety of problems and questions regarding permutation polynomials have been posed by Lidl and Mullen [3,4]. Among these there is problem of determining the number Nd of permutation polynomials of fixed degree d. In [6,9] Malvenuto and the second author address the problem of counting the permutations that move a fixed number of elements of Fq and whose permutation polynomials have ‘‘low’’ degree. Here, we consider all permutations and we want to prove the following: Theorem 1.
Let N ¼ #fs 2 SðFq Þ j @ðfs Þ5q 2g:
Then, jN ðq 1Þ!j
pffiffiffiffiffiffiffiffiffiffi q=2 2e=pq :
This confirms the common belief that almost all permutation polynomials have degree q 2. The first few values of N are listed below: q
2
3
4
5
7
8
9
11
N ðq 1Þ!
0 1
0 2
12 6
20 24
630 720
5368 5040
42 120 40 320
3 634 950 3 628 800
Proof. The proof uses exponential sums and a similar argument as the one in [2]. By extracting the coefficient of xq2 in fs ðxÞ, we obtain that the degree of fs ðxÞ is strictly smaller than q 2 if and only if X csðcÞ ¼ 0: c2Fq
For a fixed subset S of Fq , we introduce the auxiliary set of functions ( ) X NS ¼ f j f : Fq ! S; and cf ðcÞ ¼ 0 c2S
and set nS ¼ #NS . By inclusion exclusion, it is easy to check that X N¼ ð1ÞqjSj nS : S Fq
ð1Þ
550
KONYAGIN AND PAPPALARDI
Now if ep ðuÞ ¼ e2piu=p , consider the identity 0 0 11 X 1X @ X nS ¼ ep @ Trðacf ðcÞÞAA; q a2F f : F !S c2F q
q
q
which follows from the standard property ( 1 1X ep ðTrðaxÞÞ ¼ q a2F 0 q
if x ¼ 0; if x=0:
By exchanging the sum with the product, we obtain 0 1 1X@ Y X nS ¼ ep ðTrðactÞÞA: q a2F t2S c2F q
q
By isolating the term with a ¼ 0 in the external sum and noticing that the internal product does not depend on a (for a=0), we get 0 1 jSjq 1 X @ Y X nS ¼ þ ep ðTrðbtÞÞA: q q * t2S b2F a2Fq
q
Finally, nS ¼
jSjq q 1 Y X þ ep ðTrðbtÞÞ: q b2F t2S q q
Now let us insert Eq. (2) in Eq. (1) and obtain X ð1ÞqjSj Y X q1 X N jSjq ¼ ð1ÞqjSj ep ðTrðbtÞÞ: q S F q t2S S F b2F q
q
Note that inclusion–exclusion gives X ð1ÞqjSj S Fq
q
q
jSjq ¼ ðq 1Þ!
Therefore, N ðq 1Þ! ¼
Y X q1 X ð1ÞqjSj jSj ep ðTrðbtÞÞ: q S F * t2S b2Fq
q
Using the fact that for b 2 Fq* X X ep ðTrðbtÞÞ ¼ ep ðTrðbtÞÞ t2S
t 2= S
ð2Þ
ENUMERATING PERMUTATION POLYNOMIALS
551
and grouping together the term relative to S and the term relative to Fq WS, we get Y X q1 X jq 2jSjj ep ðTrðbtÞÞ : jN ðq 1Þ!j 2q S F * t2S
ð3Þ
b2Fq
q
Now let us also observe that 2 X X ep ðTrðbtÞÞ ¼ qjSj; t2S b2F q
so that 2 X X ep ðTrðbtÞÞ ¼ ðq jSjÞjSj: * t2S
b2Fq
From the fact thatQ the geometric P mean is always bounded by the arithmetic mean (i.e. ð ki¼1 jai j2 Þ1=k 1k ki¼1 jai j2 Þ, we have that 0 2 1ðq1Þ=2 Y X X X 1 ep ðTrðbtÞÞ @ ep ðTrðbtÞÞ A q 1 * * t2S t2S
b2Fq
b2Fq
ðq jSjÞjSj ðq1Þ=2 ¼ : q1
ð4Þ
Furthermore, using (3) and (4) we obtain jN ðq 1Þ!j
q1 2qðq 1Þ
X
ðq1Þ=2
jq 2jSjjððq jSjÞjSjÞðq1Þ=2 :
ð5Þ
S Fq
We want to estimate the above sum. Consider the inequality ððq jSjÞjSjÞðq1Þ=2
qq1 2
;
ð6Þ
;
ð7Þ
and the identity X S Fq
jq 2jSjj ¼ 2q
q1 ½q=2
!
552
KONYAGIN AND PAPPALARDI
which holds since
2
X
ðq 2jSjÞ ¼ 2
"½q=2 X
q
j¼0
j
S Fq ; jSjq=2
¼ 2q
"½q=2 X
! ðq jÞ
q1
!
j
j¼0
From the standard inequality ! 2n n
½q=2 X
q
j¼1
j
!
#
ðjÞ
½q=2 X
q1
j¼1
j1
!# ¼ 2q
q1 ½q=2
! :
rffiffiffi 2 22n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; p 2n þ 1=2
which can be found for example in [1], we deduce ! rffiffiffi q1 2 2q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: p q 1=2 ½q=2
ð8Þ
Therefore, (5), (6), (7) and (8) imply jN ðq 1Þ!j
q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi q 1=2 q
!rffiffiffi ðq1Þ=2 2 q qq=2 p q1
and in view of the inequalities q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi51; q 1=2 q
q q1
ðq1Þ=2 5
pffiffiffi e;
we finally obtain rffiffiffiffiffi 2e q=2 jN ðq 1Þ!j q p and this completes the proof.
&
CONCLUSION Computations suggest that a more careful estimate of the sum in (5) pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi would yield to a constant e=2p instead of 2e=p as coefficient in qq=2 in the statement of Theorem 1. However, we feel that such a minor improvement does not justify the extra work.
ENUMERATING PERMUTATION POLYNOMIALS
553
The ideas in the proof of Theorem 1 can be used to deal with the analogous problem of enumerating the permutation polynomials that have the ith coefficient equal to 0 and also to the problem of enumerating the permutation polynomials with degree less than q k (for fixed k). However, the exponential sums that need to be considered are significantly more complicated.
ACKNOWLEDGMENTS The first author was supported by Grants 02-01-00248 and 00-15-96109 from the Russian Foundation for Basic Research. The second author was partially supported by G.N.S.A.G.A. from Istituto Nazionale di Alta Matematica. The authors thank Igor Shparlinski for suggesting a substantial improvement with respect to the original result. Note added in proof. Recently the asymptotic estimate in Theorem 1 for a prime q (with slightly weaker remainder term) has been proved in the paper ‘‘The number of permutation polynomials of a given degree’’ by Pinaki Das (to appear in this journal). The author uses algebraic arguments rather than exponential sums.
REFERENCES 1. E. Grosswald, Algebraic Inequalities for p, Solution to a problem proposed by R. E. Shafer, Amer. Math. Monthly 84 (1977), 63. 2. S. Konyagin, A note on the least prime in an arithmetic progression, East J. Approx. 1 (1995), 403–418. 3. R. Lidl and G. L. Mullin, When does a polynomial over a finite field permute the elements of the field? Amer. Math. Monthly 95 (1988), 243–246. 4. R. Lidl and G. L. Mullin, When does a polynomial over a finite field permute the elements of the field? II Amer. Math. Monthly 100 (1993), 71–74. 5. R. Lidl and H. Niederreiter, ‘‘Finite Fields,’’ Encyclopedia of Mathematics and its Applications, Vol. 20, Addison–Wesley, Reading, MA, 1983. 6. C. Malvenuto and F. Pappalardi, Enumerating permutation polynomials I: Permutations with non-maximal degree, submitted. 7. C. Malvenuto and F. Pappalardi, Enumerating permutation polynomials II: k-cycles with minimal degree, in preparation.
Enumerating Permutation Polynomials over Finite Fields by Degree Sergei Konyagin1 Department of Mechanics and Mathematics, Moscow State University, Vorobjovy Gory, 119899 Moscow, Russia E-mail: [email protected],[email protected]
and Francesco Pappalardi Dipartimento Di Matematica, Universita" degli Studi Roma Tre, Largo S. L. Murialdo, 1, 1-00146 Rome, Italy E-mail: [email protected] Communicated by Wan Daging Received June 27, 2001; published online June 14, 2002
We prove an asymptotic formula for the number of permutations for which the associated permutation polynomial has degree smaller than q 2. # 2002 Elsevier Science (USA)
Let Fq be a finite field with q ¼ pf > 2 elements and let s 2 SðFq Þ be a permutation of the elements of Fq . The permutation polynomial fs of s is fs ðxÞ ¼
X
sðcÞð1 ðx cÞq1 Þ 2 Fq ½x:
c2Fq
fs has the property that fs ðaÞ ¼ sðaÞ for every a 2 Fq and this explains its name. For an account of the basic properties of permutation polynomials we refer to the book of Lidl and Niederreiter [5]. 1
To whom correspondence should be addressed. 548
1071-5797/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
ENUMERATING PERMUTATION POLYNOMIALS
549
From the definition, it follows that for every s @ðfs Þ q 2: A variety of problems and questions regarding permutation polynomials have been posed by Lidl and Mullen [3,4]. Among these there is problem of determining the number Nd of permutation polynomials of fixed degree d. In [6,9] Malvenuto and the second author address the problem of counting the permutations that move a fixed number of elements of Fq and whose permutation polynomials have ‘‘low’’ degree. Here, we consider all permutations and we want to prove the following: Theorem 1.
Let N ¼ #fs 2 SðFq Þ j @ðfs Þ5q 2g:
Then, jN ðq 1Þ!j
pffiffiffiffiffiffiffiffiffiffi q=2 2e=pq :
This confirms the common belief that almost all permutation polynomials have degree q 2. The first few values of N are listed below: q
2
3
4
5
7
8
9
11
N ðq 1Þ!
0 1
0 2
12 6
20 24
630 720
5368 5040
42 120 40 320
3 634 950 3 628 800
Proof. The proof uses exponential sums and a similar argument as the one in [2]. By extracting the coefficient of xq2 in fs ðxÞ, we obtain that the degree of fs ðxÞ is strictly smaller than q 2 if and only if X csðcÞ ¼ 0: c2Fq
For a fixed subset S of Fq , we introduce the auxiliary set of functions ( ) X NS ¼ f j f : Fq ! S; and cf ðcÞ ¼ 0 c2S
and set nS ¼ #NS . By inclusion exclusion, it is easy to check that X N¼ ð1ÞqjSj nS : S Fq
ð1Þ
550
KONYAGIN AND PAPPALARDI
Now if ep ðuÞ ¼ e2piu=p , consider the identity 0 0 11 X 1X @ X nS ¼ ep @ Trðacf ðcÞÞAA; q a2F f : F !S c2F q
q
q
which follows from the standard property ( 1 1X ep ðTrðaxÞÞ ¼ q a2F 0 q
if x ¼ 0; if x=0:
By exchanging the sum with the product, we obtain 0 1 1X@ Y X nS ¼ ep ðTrðactÞÞA: q a2F t2S c2F q
q
By isolating the term with a ¼ 0 in the external sum and noticing that the internal product does not depend on a (for a=0), we get 0 1 jSjq 1 X @ Y X nS ¼ þ ep ðTrðbtÞÞA: q q * t2S b2F a2Fq
q
Finally, nS ¼
jSjq q 1 Y X þ ep ðTrðbtÞÞ: q b2F t2S q q
Now let us insert Eq. (2) in Eq. (1) and obtain X ð1ÞqjSj Y X q1 X N jSjq ¼ ð1ÞqjSj ep ðTrðbtÞÞ: q S F q t2S S F b2F q
q
Note that inclusion–exclusion gives X ð1ÞqjSj S Fq
q
q
jSjq ¼ ðq 1Þ!
Therefore, N ðq 1Þ! ¼
Y X q1 X ð1ÞqjSj jSj ep ðTrðbtÞÞ: q S F * t2S b2Fq
q
Using the fact that for b 2 Fq* X X ep ðTrðbtÞÞ ¼ ep ðTrðbtÞÞ t2S
t 2= S
ð2Þ
ENUMERATING PERMUTATION POLYNOMIALS
551
and grouping together the term relative to S and the term relative to Fq WS, we get Y X q1 X jq 2jSjj ep ðTrðbtÞÞ : jN ðq 1Þ!j 2q S F * t2S
ð3Þ
b2Fq
q
Now let us also observe that 2 X X ep ðTrðbtÞÞ ¼ qjSj; t2S b2F q
so that 2 X X ep ðTrðbtÞÞ ¼ ðq jSjÞjSj: * t2S
b2Fq
From the fact thatQ the geometric P mean is always bounded by the arithmetic mean (i.e. ð ki¼1 jai j2 Þ1=k 1k ki¼1 jai j2 Þ, we have that 0 2 1ðq1Þ=2 Y X X X 1 ep ðTrðbtÞÞ @ ep ðTrðbtÞÞ A q 1 * * t2S t2S
b2Fq
b2Fq
ðq jSjÞjSj ðq1Þ=2 ¼ : q1
ð4Þ
Furthermore, using (3) and (4) we obtain jN ðq 1Þ!j
q1 2qðq 1Þ
X
ðq1Þ=2
jq 2jSjjððq jSjÞjSjÞðq1Þ=2 :
ð5Þ
S Fq
We want to estimate the above sum. Consider the inequality ððq jSjÞjSjÞðq1Þ=2
qq1 2
;
ð6Þ
;
ð7Þ
and the identity X S Fq
jq 2jSjj ¼ 2q
q1 ½q=2
!
552
KONYAGIN AND PAPPALARDI
which holds since
2
X
ðq 2jSjÞ ¼ 2
"½q=2 X
q
j¼0
j
S Fq ; jSjq=2
¼ 2q
"½q=2 X
! ðq jÞ
q1
!
j
j¼0
From the standard inequality ! 2n n
½q=2 X
q
j¼1
j
!
#
ðjÞ
½q=2 X
q1
j¼1
j1
!# ¼ 2q
q1 ½q=2
! :
rffiffiffi 2 22n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; p 2n þ 1=2
which can be found for example in [1], we deduce ! rffiffiffi q1 2 2q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: p q 1=2 ½q=2
ð8Þ
Therefore, (5), (6), (7) and (8) imply jN ðq 1Þ!j
q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi q 1=2 q
!rffiffiffi ðq1Þ=2 2 q qq=2 p q1
and in view of the inequalities q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi51; q 1=2 q
q q1
ðq1Þ=2 5
pffiffiffi e;
we finally obtain rffiffiffiffiffi 2e q=2 jN ðq 1Þ!j q p and this completes the proof.
&
CONCLUSION Computations suggest that a more careful estimate of the sum in (5) pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi would yield to a constant e=2p instead of 2e=p as coefficient in qq=2 in the statement of Theorem 1. However, we feel that such a minor improvement does not justify the extra work.
ENUMERATING PERMUTATION POLYNOMIALS
553
The ideas in the proof of Theorem 1 can be used to deal with the analogous problem of enumerating the permutation polynomials that have the ith coefficient equal to 0 and also to the problem of enumerating the permutation polynomials with degree less than q k (for fixed k). However, the exponential sums that need to be considered are significantly more complicated.
ACKNOWLEDGMENTS The first author was supported by Grants 02-01-00248 and 00-15-96109 from the Russian Foundation for Basic Research. The second author was partially supported by G.N.S.A.G.A. from Istituto Nazionale di Alta Matematica. The authors thank Igor Shparlinski for suggesting a substantial improvement with respect to the original result. Note added in proof. Recently the asymptotic estimate in Theorem 1 for a prime q (with slightly weaker remainder term) has been proved in the paper ‘‘The number of permutation polynomials of a given degree’’ by Pinaki Das (to appear in this journal). The author uses algebraic arguments rather than exponential sums.
REFERENCES 1. E. Grosswald, Algebraic Inequalities for p, Solution to a problem proposed by R. E. Shafer, Amer. Math. Monthly 84 (1977), 63. 2. S. Konyagin, A note on the least prime in an arithmetic progression, East J. Approx. 1 (1995), 403–418. 3. R. Lidl and G. L. Mullin, When does a polynomial over a finite field permute the elements of the field? Amer. Math. Monthly 95 (1988), 243–246. 4. R. Lidl and G. L. Mullin, When does a polynomial over a finite field permute the elements of the field? II Amer. Math. Monthly 100 (1993), 71–74. 5. R. Lidl and H. Niederreiter, ‘‘Finite Fields,’’ Encyclopedia of Mathematics and its Applications, Vol. 20, Addison–Wesley, Reading, MA, 1983. 6. C. Malvenuto and F. Pappalardi, Enumerating permutation polynomials I: Permutations with non-maximal degree, submitted. 7. C. Malvenuto and F. Pappalardi, Enumerating permutation polynomials II: k-cycles with minimal degree, in preparation.