535
Internat. J. Math. & Math. Sci.
Vol. i0, No. 3 (1987) 535-544
ON PERMUTATION POLYNOMIALS OVER FINITE FIELDS
R.A. MOLLIN and C. SMALL
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta, Canada, T2N IN4
Department of Mathematics and Statistics
Queen’s University
Kingston, Ontario, Canada K7L 3N6
(Received July 31, 1986 and in revised form October 3, 1986)
ABSTRACT.
A polynomial
if
poIFnomial
f
finite
over a
the mapping
F
F
F
field
consider the problem of characterizing permutation polynomials;
conditions
for it to represent a
of
conjecture
permutat,on
that
seek
we
is,
coefficients of a polynomial which are necessary and sufficient
the
on
a
In this paper we
is one-to-one.
f
defined by
called
is
We
permutation.
Carlitz
which
also
give
some
bearing
results
essentially that for any even integer
says
cardinality of finite fields admitting permutation polynomials of
degree
on
a
m,
the
m
is
bounded.
KEY WORDS AND PHRASES.
Polynomials, Irreduclbilit F, factorization, distribution of
values.
1980 llathematlcs Subject Classification.
I.
INTROION.
A polynomial
elements,
is
f(x)
called
known
f(a)
see
(eg.
where
GF(q)
where
GF(q) [x],
permtatlon
a
one-to-one; i.e. the
well
lIT06.
a
polynomial
q.
given a polynomial
conditions on the coefficients
d
q
since
f
f(x)
a0,al,
y-
a. x
there
is
a
d
i
1
for
surprisingly
(e.g.
little
see
with
a’s.
is given by
q
f
is
It
is
a unique
difficult
to
what are necessary and sufficient
f
to be a permutation?
has a unique such representation).
literature on permutation polynomials
references),
field
GF(q)
GF(q)
A natural question to ask (albeit
i=0
assume
finite
are a permutation of the
GF(q)
d
is"
a
the mapping defined by
[I]) that any mapping
polynomial of degree less than
answer)
is
if
[I]
which
for
deals
an
(We
may
Despite an extensive
extensive
list
of
with the classification
problem as posed above. However some very special cases are known. For example the
are
I)
cases d < 2 are trivial since polynomials of degree 0 (respectively
never (respectively always) permutation polynomials; whereas in the quadratic case
2
0), f is a permutation polynomial on GF(q) if and only
ax + bx + c (a
f(x)
2. The latter is a trivial consequence of one of the
0 and char GF(’q)
if b
R.A. MOLLIN AND C. SMALL
536
results in this paper (see Corollary 2.3 below).
has
There
been
also
Z (k/(k-.))
j:O
ntural nber.
g.c.d. (k,q
Some
of
is a permutation
gk(x,a)
<_ t < 7
with
p
-=
GF(q
of
is a permutation of
f(x)
where
GF(p),
0
b
I,
+ x
n
m+
As
g(x)
form
the
x
I.
r
if and only if
2 (mod 5),
+/-
for
f(x)
with
polynomials
considered
g(x)
GF(7 n)
but not
GF(7)
is suitably chosen then
but not on
GF(q)
GF(q)
N,
E
the
,
pp. 355-357,
if
and
only
if
x
p
+
5c
-=
x
4
+ 3x
is
shows that if
permutation
a
2m +
q
and
is a permutation polynomial on
example, S. Chowla [5]
+ bx 2 + cx, and proved that
final
3
a
5
a
is
and
Carlitz [3], in an attempt to
odd.
t
mid
generalize Dickson’s result. [4] that the quartic
For"
k
and
specialized results toward our characterization problem have also
For" example S.R. Cavior [2] investigated octics of the form
been established.
t
8
x + ax
f(x)
a E GF(q)
form
I.
I)
other-
polynomial
GF(q)
a
Dickson
the
of
0
where
so-called
For such a description the reder is referred to
where it is proved that.
2
the
polynomials
(-a)
J
of
characterization
a
those
i.e.,
polynomials
gk(x,a)
such
ax
sufficiently
2
with
prime
large
(mod p).
a
paper we completely settle the characterization problem for at least
(x)
(x). We prove that
one class of polynomials" viz., cyclotomic polynomials
m
m
In
this
is a permutation polynomial on
and
are powers of
m
GF(q)
if and only if either
m
2,
or
both
q
2.
Dickson (e.g. see [6] or [7]) classified all permutation polynomials of degree
In fact, Dickson (eg. see [I] p. 349] where it is called
Hermite’s condition since Dickson [6], [7] attributed the prime field case to
Hermite) gave necessary and sufficient conditions for a pol)qaomial to be a
permutation polynomial. However the characterization is not explicit in the sense
less
6 over
than
above,
outlined
GF[q].
and
Dickson’s
theorem
is
easy to use in the sense that is
not
extremely difficult to extract results from the theorem which give concrete criteria
in terms of the polynomial’s coefficients.
to
provide
One of the major tasks of this paper
For example
+ bx J + c GF(q)
certain polynomials to be permutations.
precisely when
glance
terms of
a,
b
and
is
and sufficient conditions, in terms of the coefficients, for
necessary
f(x)
ex
we are able to determine
at
a
is a permutation polynomial in
c.
There is a relative scarcity of permutation polynomials as may be seen by the
fact that they are necessarily polynomials with exactly one root. However this
obvious
necessary
condition
is
far
from
sufficient
as
the
example
2
3(x) x + x + over GF(3) illustrates. Another reason for the scarcity of
permutation polynomials may be seen
GF(q)
GF(q),
only
q!
by
the
fact
that,
of them are permutations.
of
the
The fact that
qq
lim
q
mappings
q!/gq
means therefore that permutations get increasingly scarce in large fields, and
points to the difficulty in our characterization problem.
0
this
A second major goal of this paper (the first being progress in the
characterization problem) is to make advances in establishing Carlitz’s conjecture,
PERMUTATION POLYNOMIALS OVER FINITE FIELDS
namely
for a given even integer
that
m,
q
may be chosen sufficiently large such
that there are no permutation polynomials of degree
m
[8]
Hayes
GF(q).
on
conjecture when the characteristic of the field does not divide m.
the
established
537
However he used the deep Lang-Weil Theorem [9] which is closely cohnected to the
Rieman,
hy-,thesls for curves over finite fields (eg. see [I, p. 331] for an
Our tec[uniques and .thod of proof
exqlanation of the connection).
less
are
complicated.
GF(q)
over
degree
exist
this
in
of a certain polynomial, no permutation polynaials of any given even
on GF(q) for sufficiently large q. Of course there are infinitely
GF(q)
many permutation polynomials of a given odd degree on infinitely many
will
be
seen
in term of
paper
We establish that, subject to the absolute irreducibility
f(x) :mx
by our characterization of such polynomials as
a,
and
b
%2).
(see
c
The
latter
as
J
+ bx + c
work
the
generalizes
of
I0].
Niederreiter and Robinson
conjecture follows from one of our main results
provlded that any one of several conditions (which we will outline) can be shown to
We
note
It
should
that
Carlitz’s
hold.
monograph
],
connection
with
(that is,
the
be
the
finite
rich
a
source
that
F)
of
groups,
it
was
is
polynomials
permutation
their
The linear groups over finite fields
full
linear
group of all
F
linear
invertible
were already well known (at least in the prime-field case)
simple
liberation of groups from their
matrix
1896 thesis [4] and his 1901
Dickson’s
studying
groups.
the
of
both
in
for
simple
subgroups
transformations over
ms
noted
motivation
groups.
Indeed,
representations
while this may have delayed the
groups of substitutions and
as
doubtless the route by which, primarily thanks to Dickson,
the prime
finite fields entered mathematics in a serious way, except of course for
fields, which had arrived on the scene centuries earlier via number theory.
Dickson
wanted to know which polynomials represented permutations because he wanted to study
linear groups (which are after all groups of permutations) by computations with
applications
to the linear groups, and his methods did lead him to previously unknown classes of
simple groups (see [4], part II, 17, as well as pert II of [6]).
generators for them.
Finally, the reader is referred to the well-written
Niederreiter
1,
the
The context, in both [4] and [6], is always
7
Chapter
for
an
outline
of
compendium
basic
results
polynomials, and to Schmidt [II] for some results on equations
over
by
Lidl
and
on permutation
finite
fields
which we will need in %3.
2.
THE CHARACTERIZATION PROBLEM.
We begin by observing that the composition of two polynomials is a permutation
polynomial if and only if each constituent is one. This leads to the following
useful fact.
LE 2.1.
Let
f(x)
m.
n
i
E c.x
i
i=l
GF(q)
where
m
n
m
m
n-I
n
and
i:l
on
GF(q)
c.
0.
Suppose
e
g.c.d.
l<i.<n
if and only if
g.c.d. (e,q
{mi}.
I)
Then
I
f(x)
and
is a permutation polynomial
m./e
n
1
E c.x
1
is
a
permutation
R.A. MOLLIN AND C. SMALL
538
polynomial.
only if
result
following
x
The proof is immediate
is a permutation polynnial of
if
GF(q)
and
I, (see [I, Theorem 7.8 (ii), p. 351]).
g.c.d. (e,q- I)
The
[I0, Lemma 5, p. 210].
generalizes
e
observation
This
from the fact that the monomial
characterizes
polynomials
of
class
certain
a
as
permutation polynomials in terms of their coefficients.
THEOREM 2.2.
j
Suppose that k and
are positive integers such that
I)
.j >_
and g.c.d. (k
I. Then ax k + bx o + c with a 0 is
k
j,q
q
a permutation of
PROOF.
x
k +
bx 0
+
ax
a- IbxJ
if and only if
GF(q)
k
+ c
is
a
permutation
pol,omial
if
-lb.
a
f(x)
g.c.d. (k
ax J
x
k
is
then
I)
j,q
Therefore
f(x)
x
j(xk-j
a
u
I.
permutation
polynomial
GF(q )k-j
say
x j(x
a)
If
k-j
u
yk-j),
0.
b
and
-a
Let a
g.c.d. (k,q
I)
is a permutation polynomial.
a permutation polynomial if and only if
and
I)
g.c.d. (k,q
and
0
only
k
x
Assume that
0
Since
0.
f(0),
a
Q.E.D.
contradiction.
COROLLARY 2.3.
if
0
y
with
f(y)
so
is
#
GF(q).
on
y
if
then
and only if
ax
+ bx +
0)
(a
c
The following advances Cavior
GF(q)
is a permutation polynomial on
and the characteristic of
0
b
2
GF(q)
is
2.
2 ].
is not divisible by 3, 5 or 7. Then
Suppose
q- I
t < 8, is a permutation polynomial on GF(q)
if and
0 and GF(q) has characteristic 2.
only if a
g.c.d. (8- t,q- I)
PRDOF.
Therefore Theorem 2.2
by hypothesis.
0 and g.c.d. (8,q
I; i.e., q must be even.
I)
applies and forces a
COROLLARY
x
8
+ ax t
2.4.
t
for
odd and
@.E.D.
is necessary for
Note that the restriction q
modulo 3, 5, or
7
8
Corollary 2.4 to hold since it is known for example that x + 4x permutes GF(29),
5
8
3
8
Moreover it is open as to whether x + x
and
GF(II).
x + ax
permutes
permutes GF(13 n) or GF(7 n) for odd n. For details see Cavior [2].
In the next section we will make headway on the question left at the end
Cavior’s
paper,
of degree
We
2k
on
note
conjectured to be true by Carlitz, namely that for a given
and
there exists a bound
N
k
GF(q),
furthermore
there is no permutation
N
k
4 as above.
in particular for k
such that if
that if we z-emove the condition
j)th
power
f()
f(0)
GF(13),
f(5)
0.
yet
f(-3)
g.c.d. (k-
must
m
I)
for
new
0,
GF(q). One simple observation is that
cannot be a permutation polynomial unless a fails to be. a
@k-j
for some ] m GF(q) then
for if
a
GF(q)"
However this is not a sufficient condition" 3 is not a cube in
4
since
GF(13)
f(x)
3x
is not a permutation of
x
in
I.
Furthermore, Dickson [4, p. 77] (see also [I]), proved that
degree
,q-
search
to be a permutation of
criteria for
f(x)
j
k
ax
f(x)
a
x
(k
we
k
polynomial
q
from the hypothesis of Theorem 2.2 then, as noted above,
of
over
GF(q)
cannot permute
immediately gets from this discussion"
GF(q)
if
q
a
(mod m).
polynomial
of
Therefore one
PERMUTATION POLYNOMIALS OVER FINITE FIELDS
a # 0
THEOREM 2.5.
-I
eund -ha
nd only if
b
Let
f(x)
0
in
ax
+ bx
th
{k
j)
GF(q)
and
is a
k
j
+
539
be a polynomial over
c
power in
Then
GF(q).
g.c.d. (k,q
GF(q)
with
permutes GF(q) if
f
I.
i)
-ha -I
j)th power is further
to fail to be a (k
4
demonstrated by the following example" x + 3x permutes GF(7) although 3 $ 0
in GF(7) and g.c.d. (4,6)
I.
2
The clear necessity for
An immediate corollary of Theorem 2.5 is"
COROLLARY 2.6
If f is as in Theorem 2.5 and -ha -I
where
d
g.c.d. (q
GF(q)
j) then f permutes
l,k
b
0 and g.c.d. (k,q
1)
1.
Theorem
g.c.d. (k
k
j
2.2
other
divides
with
+
a
for
0
and
is a
ax
x
Since
Thus
GF(q).
k’-I
k’-2 y
Thus
x
k’
y
g.c.d,
0
a(x
contradiction.
I)
q
GF(q),
I.
$
@
if and only if
x
k
ax
permutes
f
then
if and only if
x
GF(q)
I.
j
if and only if
permutes
k/j
ax
GF(q)
if and only if
GF(q)
then by Le,a 2.1
0)
$
b,
a,
and g.c.d. (j,q
I)
(k/j)-I
z
+ z (k/j)-2 +
permutes
f
Then
(equivalently a
.
d,
where
k;
divides
j
permutes
-I
have
we
GF(q).
Let
Yk’-I
dth
k’-I
y
where
c
1
0
GF(q)
+ yk’-I
k’
$
is a
a
+
+ x
b
permutes
E
x
If
z
b
If
-I
+ bx j +
GF(q)
permutes
1.
j
k
GF(q),
I.
I)
z
-I.
1)
ax
g.c.d. ((k/j)
th
d
power in
some
-ha
a
k/j
0;
$
f(x)
g.c.d. (k,q
k
that
x
y
f(x)
g.c.d. (k,q
F.
were
Let
-ba-l-I
Suppose
k’
condition
is odd and
k.
GF(q)
b
in
condition (replacing it
thereby allowing us to make headway in such cases where j
conditions)
THEOREM 27.
c
power
if and only if
unsatisfying
that
in
the
does not allow us to touch the case where q
is even. The following result relaxes the
with
GF(q)
somewhat
is
I)
j,q
th
d
is a
for some
power in GF(q) we have t@
I. Then
y since z
yz where x
Now let x
k’-I
k’-2
k’-I k’-I
a.
y
1)
z
+
+
+
y
+ z
(z
k’
k’
sx with x $ y, a
y
ay : x
and so
y)
Q.E.D.
We note that as a straightforward application the reader may use Theorem 2.7
degree
3.
This
requires reducing
of
to
characterize
polynomials
3
3
2
x. If a is a square there is no
f(x)
ax + bx + cx + d to the form x
2
2
x + xy + y
When a is not a square one shows it can be written as a
problem.
2
is a non-square.
for which z + z +
with x $ y by finding an element z $
The result is Corollary 2.9 below.
However
2.7
Theorem
may
be
to
difficult
technical condition must be satisfied.
However when
apply in general since a rather
k
j
2
we have
a
simpler
0
and
k > 2)
result.
THEORE
permutes
2.8.
GF(q)
where
a
-I
-ha
Thus we assne
n
(q
+/-
then either
As before,
PRf)OF.
f(x)
If
f
l)/k.
and the fact that
We
is
It
f
a
f
k
ax
+ bxk-2 +
q m +I (mod k)
permutes
(where
c
b
or
if and only if
GF(q)
a
0.
x
k
-ax
k-2
does,
assume the conclusion is false and reach a contradiction
is clear that
n
is a permutation we
+
q
permutation,
q
I.
have
(mod
k
and
a
0.
Let
Hence, using [i, Lemma 7.3, p. 349]
r.
xGF q
(xk-oxk-2) n
R.A. MOLLIN AND C. SMALL
540
therefore"
0
-a
Z
Z
this
only for
occur
can
0
i
n
kn-2i
x
xGF q
(when nk
(-) i
In the former case we get
(-a)
Z
i=O
x
xGF(q) i=O
by the same lemma from [I] we have
unless
0
q- I)
But
q + I).
contradicting
E
x
Z
xGF (q)
xGF (q)
i:O
I.
nk
(when
i
or
Now
x
xGF(q)
kn
2i
q
Q.E.D.
[I, ibid], and similarly in the remaining case. This completes the proof.
have characteristic different from 3. Then
Let
GF(q)
COROLLARY 2.9.
2
3
2
3ac
and
ax + bx + cx + d (a # O) permutes GF(q) if and only if b
f(x)
2 (mod 3).
q
+ ba-lx + ca-1) does. Put
y3 + c’y + d’
with
ca1,
ba-lx
+
.x (x +
then
y
x + b(3a)
3
b2)(3a2) -I. Thus f permutes GF(q) if and only if x ax does
(3ac
c’
2
3ac)(3a2) -1 Observe that a 0 means b 2 3ac. By Theorem
(b
where a
x(x
if and only if
permutes GF(q)
f
PRfX)F.
2
2
2.8,
a # 0
if
perg,utation
is not a permutation polynomial.
f
then
I,
1,3)
g.c.d. (q
polynomial if and only if
If
is a
2 (mod 3).
q
i.e.,
f
then
0
a
Q.E.D.
Now we consider the case where the degree of the polynomial is a power of
the
characteristic.
PROPOSITION
p
Let
2.10.
GF(q)
0. Then"
a
GF(q) and s
l)th power in
permutes GF(q) if and only if a is not a (pS
I)
I, 8xl
I, pS
If f permutes GF(q) then g.c d. (q
is a primitive root in GF(q) then g.c.d. (q
l,ps- I)
If
only if f permutes GF(q).
x
f(x)
(I)
ux
0
with
If
PRfF.
has both
0
a
and
as roots, hence
0
@
th
(pS
is a
S
I
a
#
c
c2
If
since
)ps-I
GF(q).
If
i.
GF (q),
power in
not
is
a
say
f
then
@
Conversely
permutation.
if
S
S
c2P
ac
ps-1,
a
and
if
ac
then
2
c2)P
(c
a(c
c2),
and
This proves (I)
s
g.c.d. (q
aq-I
only if
that c P
I
such
2
(c
let
$
f
(2)
(3)
c
and
p,
characteristic
have
S
l,p
I)
then
This secures (2).
s
l,p
g.c.d. (q
a(q-l)/g
I.
I)
However a
g
is a
a
then
1
l)th
(pS
is a
Thus,
is primitive.
a
power
l)th
(pS
is not
in
GF(q)
power if and
(pS
a
l)th
@. E. D.
power.
We
note that an immediate consequence of Proposition 2.10 is [I0, Theorem I0,
p. 209 ].
we
Finally
polynomials
@
m
close
(x)
2.1 I.
polynomial on
of
2.
GF(q)
this
section
with
a
characterization
of
cyclotomic
which are permutations.
The
m
th
cyclotomic
if and only if either
m
polynomial
2,
or both
m
m
(x)
and
is a permutation
q
are
powers
PERMUTATION POLYNOMIALS OVER FINITE FIELDS
POOF
If
2
m
a
polynomial if and only if
then
x
and
permutation
polynomial,
g.c.d. (2,q
I)
I;
i.e.,
(i)"
If
m
CASE
CASE (ii)
b,
for
a
2
+
then
a
it
Conversely if
m
the
twice
a permutation
which
is
2a-I
x
is
one
if
and
x
is
m
(I)
of
power
@
m
a
only
if
(0);
odd
an
always
then:
2
is not a power of
m
an odd prime, then
not
is
m
x
If
p
(-I)
(x)
is one.
2.
p
2p
If
m
3.
2a-I
m
541
a-I
and
then
prime
Q.E.D.
(0).
CARLITZ CONJE AND E%N DEGREE PERKrATIONS.
We begin with a characterization of permutatation polynomials in terms of
solutions of certain
0.
f(x,y)
The equivalence of (I) and (2) in what follows is a natural generalization
of
[I0, Lemm 4, p. 210], whereas the equivalence of (2) and (3) is the central idea of
Hayes [8].
We
later use this result to make headway towards the Carlitz conjectu’e
will
mentioned previously.
Let
THEOREM 3 i.
n
i
Z c. x
I
i=l
f(x)
GF(q)[x]
E
with
c
0.
n
Then
the
following are equivalent:
(I)
f(x)
(2)
The
is a permutation polynomial on GF(q);
n
-I
n-i
Z c
equation
+
+
n ciY
i=l
xi-I xi-2
GF(q) x GF(q),
(x0,Y0)
(3)
The
(x0,Y0)
PIF.
GF(q) x GF(q)
First
we
with
f(a)
loss of generality that
0.
n
Then
b
n
i
Z c .b [(ab-
i=l
i=iCiYoE
-i
y
0.
f(b)
0.
and
Conversel.r, suppose that
and
0
has
with
It follows that
-I
n-i
n
Z c
i
i:l
x0Y0
Cn cyn
This establishes the equivalence of
b,
a
Now, let
-I
hencei=lE Cn
n
Z
i=l
solutions
no
Y0"
n
O,
1]
0;
Y0
y)
x0
I]
I
Ix 0 i
or
0
only has solutions
0
the equivalence of (I) and (2).
prove
permutation polynomial then
Thus
x
f(y) )/(x
(f(x)
equation
E
with either
+ I]
x
CiYO
i-I
[x 0
-I i
and we
,y
ab
0
n-i
Ix0
is not a
assume
without
aKl
i
+ x n,,,i-2 +...+1]
n
ric i (Y0
-I i
b
Y0
O,
for
with
x0Y0
2 ), and the equivalence of
xr,
-I
2
Y0
and
is clear.
m
g.c.d.
f(x,y)
-i
3
.E.D.
THEOREM
0
0
O.
1)
ll/Ix 0
i=
and
f(x)
If
m
Let
3.2.
ran,
2
g
q
Z c
n
i:l
c_y
1
with
I.
n
Put
0
c.
GF(q)
g.c.d. {m i}
for
g
1
i
and
,n;
assume
<_ i_<_n
Suppose that
1.
n
m.
n
1
Z c.x
i:l
f(x)
x
+ x
+
+
is absolutely irreducible
542
R.A. HOLLIN AND C. SMALL
GF(q),
over
m.’
1
where
Then, whenever
mi/g.
permutation polynomial over
250(m
q
1)
n
5
f(x)
is not a
GF(q}.
if and
By Lemma 2.1, f(x} is a permutation polynomial on GF(q)
m.’
n
1
is one. Now let N be the number of zeros of f(x,y) in
only if
Z c.x
i
i:l
5/2 1/2
I)
q
N
GF(q) x GF(q). By [II, Theorem IA, p. 92] we have
ql j(mn
PROOF.
Now
N
let
$
mn’-mi’ m
n
Z c
n
i=l
we have
x0
the number of zeros of
be
ciY0
polynomial of degree
Y0
0
n
m "-I
n
have
we
x0
values for
N
2(m
be
a
n
0
with
Since the
m
and there are at most
1
m
i
f(x,y)
m ’-2
n
+
x0
+
+
..... I)5/2
I)
N
since
q
J(mn
unless
polynomial
permutation
N
q
1/2
N*.
N
are
i
$
or
0
$
0
n
this
and again there
n’
2(m
I)
and by Theorem 3.1
Hence
f(x)
ex’e
For
0.
Y0
\’alues for
m
0,
In total then we have
x0
x
is
a
If
Y0’
at ,st
But
clearly
f x
cannot
is not a permutation
Q.E.D.
polynomial on OF(q).
Note that as special cases of Theorem 3.2 we recover Theorem 9,
Lemm
7
and
Theorem II of [I0].
Now we state one final result which links the results of %2 and %3.
k
fX)ROLLARY 3.3.
ax + bx + c (a $ 0) with k not a power of
Let
f(x)
Then
f permutes
1)
250 (k
the characteristic of GF(q), and suppose q
I.
0 and g.c.d. (k,q- I)
GF(q) if and only if b
of Theorem 3.2 is absolutely
f(x,y)
By
PROOF.
I0, Lemma 3, p. 208
irreducible in this case. Thus by Theorem 3.2 f is not a permutation polynomial
5.
GF(q)
on
b
unless
polynomial if and only if
to
q
0.
But once we know b
g.c.d. (k,q- I)
0
then
f
is a permutation
I, as before.
250 (k
I)
In fact, the hypothesis q
see [I0, Theorem 9].
> (k2 4k +
5
Q.E.D.
in Corollary 3.3 can
be
weakened
6)2;
AEMENT.
The research for this paper was supported by N.S.E.R.C. Canada.
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1.
LIDL,
2.
CAVIOR, S., A Note on Octie Permutation Polynomials, Math.
450-452.
3.
CARLITZ, L., Permutations in Finite Fields, Acta Sci. Math. Szeded, 2._4 (1963),
196-203.
4.
DICKSON, L.E., The Analytic Representation of Substitutions, Ann. of Math.,
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I__I
5.
CHOWLA, S., On Substitution Polynomials (mod p), Norske Vid. Selsk. Fordh.,
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4./1
R. and
(1983).
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(1963)
PERMUTATION POLYNOMIALS OVER FINITE FIELDS
543,
6.
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7.
DICKSON, L.E., Linear Groups, Dover, New York, (1958).
8.
HAYES,
9.
LANG, S. and WEIL, A., Number of Points of Varieties in Finite Fields, Amer. J.
Math., 7_6 (1953), 819-827.
the
Galois
Field
’rheory,
D.R., A Geometric Approach to Permutation Polynomials over a Finite
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I0.
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II.
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