JOURNAL OF COMBINATORIALTHEORY7, 49-55 (1969)
The Degrees of Permutation Polynomials
over Finite Fields*
CHARLES WELLS
Case Western Reserve University, Cleveland, Ohio 44106
Communicated by Alan J. Hoffman
Received January 10, 1968
ABSTRACT
A number of theorems are proved concerning the connection between the cycle
structure of a permutation of a finite field GF(q) and the degree of the polynomial
representing it. In particular (Section 4) if Kr is the set of permutations of GF(q)
moving ~< r elements, then, if r grows slowly enough with respect to q as q --~ 0%
almost all polynomials of degree < q -- 1 representing permutations in Kr have
degree q -- 2.
1. INTRODUCTION
Let r be a permutation of the elements of the finite field GF(q = pn).
It is well known that there is a unique polynomial of degree ~ q -- 2
which r e p r e s e n t s f in the sense that for all a ~ GF(q), f ( a ) = r
Such
a polynomial is called a p e r m u t a t i o n p o l y n o m i a l . In this paper some
results on the degrees of permutation polynomials are presented. Each of
the theorems says roughly that, if K is a class of permutations which move
a number of elements of GF(q) which is small compared to q, then, as
q ~ ~ , the number of permutations in K represented by polynomials of
degree q -- 2 is asymptotic to the number of permutations in K.
2. SOME SPECIAL RESULTS
Let a, b ~ GF(q), a ~ b. It may be verified by direct substitution, using
the fact that, if r ~ 0 in GF(q), then r q-1 ---- 1, that the polynomial
f ( a ) = x + (a - - b ) ( x - - a) q-1 - k (b - - a ) ( x - - b) ~-1
* Supported by NSF Grant GP 6565.
49
58a171 I-4
(1)
50
WELLS
represents the transposition (a b). Then, as in the case of any permutation
polynomial [3, p. 59], the coefficient of x q-1 is zero, so d e g f ~< q -- 2.
But it is not hard to show that the coefficient of x q-2 is (a -- b) 2 =/= 0, so
that every transposition in GF(q) is represented by a unique polynomial
of degree q -- 2.
If (a b c) is a 3-cycle o f elements of GF(q), it is represented by
g ( x ) - - x + ( a - - b ) ( x - - a) q-1 @ ( b - - c ) ( x
- - b) q-1 -]- ( c - - a ) ( x - - c) q-1 .
(2)
The coefficient of x ~1-2 here is
)t - - a ( a - - b) -]- b ( b - - c) + c ( c - - a),
so that ,~ = 0 if and only if a is a solution o f the quadratic
x 2-(b+c)
x+b
2+c
2-bc
= O.
(3)
This has discriminant
D = - - 3 ( b - - c ) ~,
which has a square root in GF(q) if and only if --3 is a square in GF(q).
We first consider the case in which q is odd. In this case - - 3 is a square
in G F ( p ) if and only if the Legendre symbol (--3)/p -- t. But by the
quadratic reciprocity law, if p = 1 (mod 4), then
and (p/3) ~ 1 if and only if p -- 1 ( m o d 3). If p -~ 3 (rood 4), then
yielding the same result. If - - 3 is a square in GF(p), it is a square in G F ( p ~)
for any k. If - - 3 is not a square in GF(p), then it is a square in G F ( p ~) if and
only if k is even. But - - 3 is not a square in GF(p) if and only i f p -- 2
(mod 3). Since in this case k is even if and only ifp k -- 1 (mod 3), we have
that, when p @ 2, p @ 3, then --3 is not a square in GF(q) if and only
if q ~ 2 (mod 3).
W h e n q ~ 1 (mod 3), the solutions o f (3) are
x =
89
where r 2 = --3. The solutions are easily seen to be distinct from each
DEGREES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
51
other and from b and c. So for every one o f the q(q -- 1) ways of choosing
b and c there are exactly two a's for which (a b c) is a 3-cycle represented
by a polynomial of degree < q -- 2.
N o w suppose q is even. We m a y write (3) here as
(x § b)(x + c) = (b + c) 2 ;
(Y)
setting s = b + c and y = s-l(x q:- b), this is y2 + y _? 1 = 0, which is
irreducible over GF(2 ~) if and only if n is odd. Again, the roots o f (3'),
when they exist, are distinct from each other and f r o m b and c, so precisely
the same result holds in this case. (The even case is due to D. Hayes.)
Since (a b c), (b c a), and (c a b) are all the same, this proves
THEOREM. Every transposition over GF(q) is represented by a unique
polynomial of degree q - 2. I f q ..... 2 (mod 3), then every 3-cycle is
represented by a polynomial of degree q -- 2. I f q -- 1 (mod 3), then all
but ~q(q
1) 3-cycles are represented by polynomials of degree q -- 2.
The connection with the comments at the end of the first section is
clear when it is noted that there are 3q
1 (q _
1)(q -- 2) 3-cycles of elements
of GF(q) altogether.
The preceding two theorems have the following corollary:
COROLLARY. The symmetric group of permutations of GF(q) is generated
by the permutations represented by polynomials of degree q -- 2. I f q ...... 2
(rood 3), then the alternating group is generated by the even permutations
represented by polynomials of degree q -- 2.
In this connection the results of [6] should be noted.
If q -- 3", one m a y prove similarly that all but 3'~(3~ -- 1) 3-cycles are
represented by polynomials of degree q -- 2. Certain other special results
may be obtained. For example, an explicit formula m a y be obtained
for products o f two disjoint transpositions. But even for 4-cycles an
explicit formula seems difficult to get. The procedure used above to
obtain the formula for 3-cycles breaks down because, when the polynomial
corresponding to (3) has roots, the roots m a y not be distinct from the
other elements already chosen to be in the 4-cycle (this happens already
in GF(25)). However, it is true that a polynomial representing a 4-cycle
always has degree q -- 2 if q - - 3 (rood 4).
One m a y also treat in the same way the question o f whether or not
other coefficients occurring in the representing polynomial are zero. F o r
example, consider again the transposition (a b), a ~ b. In general, the
52
WELLS
coefficient o f x '~-x ~ (for 0 ~ r ~ q -- 1, i" @ q - 2) in the polynomial
representing (a b) is
;'_
){a --
__
However,
(qq-I
1- r)
(-- I)" (mod p).
(4)
This is clear if r :,: 1. It is well k n o w n and easy to show that
(q
--1
q-- 1
1)'4-(qq-.... I"-- ,
--1
1 r) = (q
. . . .
q
r)
1-"
But the right side is 0 (rood p) by the Lucas criterion (this m a y also be
shown by induction). Then (4) follows by induction on r.
It follows from (4) that the coefficient o f x ~- 1 ~ is simply ( a - b)
(a r -- b r) for r ~- q -- 2, and the coefficient o f x is (a -- b)(a ~t-2 - - b q 2) + 1.
Results analogous to the preceding results on degrees can be read off
from these formulas. F o r example, excluding the anomalous case
r == q -- 2, there are a and b in GF(q) for which the coefficient o f x q-a-~
in the polynomial representing (a b) is zero if and only if (r, q -- 1) 7-A 1.
The coefficient o f x is more complicated, but it m a y be proved using the
same kind of arguments as for the case o f the 3-cycle that there are
a, b c GF(q) for which the coefficient o f x is zero if and only if one of these
two possibilities occurs: ( 1 ) p =-=:2 (here setting a -- 0 or b = 0 is sufficient,
but there are pairs (a, b) neither of which is zero which yields a zero
coefficient o f x as well); (2) p : ~ 2 and q :--~ 1 or 4 (rood 5).
3. A N ASYMPTOTIC RESULT
Let r and s b e fixed positive integers, and k,, .... , ks non-negative integers
for which
• iki ~
r.
(5)
i=2
Let P(k2 ,..., k.~) be the set o f permutations o f GF(q) which are the disjoint
products o f k., transpositions, ka 3-cycles, etc. Such permutations move
exactly r elements.
THEOREM. T h e r e is a c o n s t a n t C = C(k2 .... , k.~).for which P(k~ .... , ks)
DEGREES OF P E R M U T A T I O N P O L Y N O M I A L S OVER F I N I T E FIELDS
53
contains Cq !/(q - r) ! permutations, o f which not more than 2q !/(q -- r q- 1) !
are represented by p o l y n o m i a l s o f degree less than q -- 2.
COROLLARY. The n u m b e r o f p e r m u t a t i o n s in P(k2 ,..., k~) represented
by polynomials o f degree q - 2 is a s y m p t o t i c to the n u m b e r o f all
permutations in P(k2 .... , ks) as q -+ oo.
PROOF OF THEOREM: Let (a s ,..., G) be an arbitrary r-tuple of distinct
elements of GF(q). There are clearly q!/(q -- r)! such r-tuples. The m a p
which takes each r-tuple (a 1 ,..., a~) into the (unique) permutation in
P
P(k2 ..... ks) which contains al ..... ar in that order is not injective.
F o r example, (a, b, e) and (b, c, a) both go into (a b c). But there is a
n u m b e r B = B(k,,, ..... ks) not dependent on (al ,..., a~) with the property
that each permutation in P comes under/3 from exactly B such r-tuples.
In fact [5, p. 67]:
B = 1![ ik'(k~!)
(6)
i=2
If we n o w let C =- i / B , we have the first part of the theorem.
N o w a permutation of P(ka .... , ks) is represented by a polynomial
f(x)
- - x + y , y~
(a,.j., -
a,,.,+O(x
-
a,.~.,)~-~
(7)
i = 2 ]=1 n ~ l
(cf. (1) and (2)), where the given permutation
(i ~ 2,.., s ; y ~ 1..... k~ ; n = 1,..., i) and where
It follows that the coefficient o f x c'-~ is given by
coefficients which are polynomials in the other
Q(a~,.s,O - -
~
i=2 J=l
F, ai.~.,,(a,.~.,, -
takes a;j.,~ into ai.i,,~+a
one sets a,,5.i+~ =- ai.i.1
a quadratic in a2.a,1 with
ai.j.,,. This quadratic is
9
a,.,.,+O.
(8)
n=l
Thus an r-tuple (as ..... a~) determines a permutation in P(k,2 ,..., ks) of
degree less than q -- 2 if and only if a2.1.a (which is ax) satisfies
Q(a2,,.O = o.
There are q!/(q - - r _c 1)! possible choices for a~ ,..., a, ; once these are
chosen there are at most 2 possibilities for a l . Thus there are at most
2q!/(q -- r q 1)! permutations in P(kz .... , ks) represented by polynomials
of degree less than q -- 2.
Since the ratio o f this n u m b e r to the order o f P(k2 ..... ks) is
2/[C(q -- r + 1)], the corollary follows.
54
WELLS
4. FURTHER RESULTS
The theorem of Section 3 requires r to be fixed. It is possible to allow r
to grow slowly with respect to q and obtain the same result, as the following
theorem asserts.
Let F denote the usual F function restricted to the real numbers ~1.
Then, for any integer n > 0, F(n) = (n -- 1)!, and F i s injective. We define
the factorial root Rt(c~) of an arbitrary real number c~ >~ 1 by
Rt(~) = F - ~ ( a ) - 1.
(9)
Let N(q, r) be the number of permutations of GF(q) which move at most
r elements, and let M(q, r) be the number of such permutations represented
by polynomials of degree < q -- 2. Then we have
THEOREM. For any E > O, as q -~- o0,
M(q, Rt(ql-~)) -~ O.
N(q, Rt(ql-~))
COROLLARY. For f i x e d r, as q ~
~,
M(q, r)
-~ 0.
N(q, r)
PROOF OF THEOREM:
By the theorem of Section 3,
q~
N(q, u) = 1 4- ~ (q = r)! E Cer
r=2
(10)
Pr
where P~ ranges over the set of all P(k2 ..... k~) such that k2 ..... k~ satisfy
(5) and where C p is the constant C of the theorem of Section 3 for that
P ( k 2 ..... k~). If A~ is the number of cycle classes of permutations which
move not more than u elements and if Co = min~,P r C p , then
N(q, u) > AuCoq!/(q -- u)!.
We now estimate Co. Let Bpr = B defined by (6). We recall that Be
is the number of r-tuples yielding the same permutation of Pr and that
Be t depends only on P~, not on the particular permutation of P,..
Moreover, C e = 1/B~ r . Now if two r-tuples give the same permutation
of P~, then one must be a rearrangement of the other, so that max~.e
B e ~< u!. Hence C o >~ l/u!. Therefore
N(q, u) > Auq!/u!(q -- u)!.
DEGREES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
55
Now
M(q, u) <~ 2Auq !/(q - - u -{- 1)!
so t h a t
M(q, u)
N(q,u)
2u!
q--u+
~
1 "
I f u ~ Rt(qa-'), t h e n
M(q, u)
2q 1-~
N(q, u) <~ q - - ~
ql--E
q__ql
,
2ql
ql-,(q,_
1)
---~0
as q - § oo. T h i s p r o v e s t h e t h e o r e m .
The corollary follows by an easy adjustment of the proof of the theorem.
ACKNOWLEDG MENT
The author wishes to acknowledge the helpful suggestions of L. Carlitz and D. Hayes,
REFERENCES
The author will be happy to supply on request an extensive bibliography of works
concerned with permutations of algebraic structures.
I. L. CARLX'rZ, Permutations in Finite Fields, Acta Sci. Math. Szeged24 (1963), 196-203.
2. L. CARLITZ AND C. WELLS, The Number of Solutions of a Special System of Equations
in a Finite Field, Acta Arith. 12 (1966), 77-84.
3. L. E. DICKSON, Linear Groups with an Exposition of Galois Fiehl Theory, Dover,
New York, 1958.
4. D. HAYES, A Geometric Approach to Permutation Polynomials over a Finite Field,
Duke Math. J. 34 (1967), 293-305.
5. J. RIORDAN, An Introduction to Combinatorial Analysis, Wiley, New York, 1958.
6. C. WELLS, Generators for Groups of Permutation Polynomials, Acta Sci. klath.
Szeged 29 (1968), 167-176.