<s>
In	O
mathematics	O
,	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
(	O
for	O
a	O
given	O
ring	O
)	O
is	O
a	O
polynomial	O
that	O
acts	O
as	O
a	O
permutation	B-Algorithm
of	O
the	O
elements	O
of	O
the	O
ring	O
,	O
i.e.	O
</s>
<s>
the	O
map	O
is	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
Over	O
a	O
finite	O
field	O
,	O
every	O
function	O
,	O
so	O
in	O
particular	O
every	O
permutation	B-Algorithm
of	O
the	O
elements	O
of	O
that	O
field	O
,	O
can	O
be	O
written	O
as	O
a	O
polynomial	O
function	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
finite	O
rings	O
Z/nZ	O
,	O
such	O
polynomials	O
have	O
also	O
been	O
studied	O
and	O
applied	O
in	O
the	O
interleaver	O
component	O
of	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
algorithms	O
.	O
</s>
<s>
A	O
polynomial	O
with	O
coefficients	O
in	O
(	O
symbolically	O
written	O
as	O
)	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
of	O
if	O
the	O
function	O
from	O
to	O
itself	O
defined	O
by	O
is	O
a	O
permutation	B-Algorithm
of	O
.	O
</s>
<s>
the	O
function	O
is	O
onto	O
(	O
surjective	B-Algorithm
)	O
;	O
</s>
<s>
(	O
Hermite	O
's	O
Criterion	O
)	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
of	O
if	O
and	O
only	O
if	O
the	O
following	O
two	O
conditions	O
hold	O
:	O
</s>
<s>
If	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
defined	O
over	O
the	O
finite	O
field	O
,	O
then	O
so	O
is	O
for	O
all	O
and	O
in	O
.	O
</s>
<s>
The	O
permutation	B-Algorithm
polynomial	I-Algorithm
is	O
in	O
normalized	O
form	O
if	O
and	O
are	O
chosen	O
so	O
that	O
is	O
monic	O
,	O
and	O
(	O
provided	O
the	O
characteristic	O
does	O
not	O
divide	O
the	O
degree	O
of	O
the	O
polynomial	O
)	O
the	O
coefficient	O
of	O
is	O
0	O
.	O
</s>
<s>
There	O
are	O
many	O
open	O
questions	O
concerning	O
permutation	B-Algorithm
polynomials	I-Algorithm
defined	O
over	O
finite	O
fields	O
.	O
</s>
<s>
However	O
,	O
Dickson	O
was	O
able	O
to	O
use	O
it	O
to	O
find	O
all	O
permutation	B-Algorithm
polynomials	I-Algorithm
of	O
degree	O
at	O
most	O
five	O
over	O
all	O
finite	O
fields	O
.	O
</s>
<s>
A	O
list	O
of	O
all	O
monic	O
permutation	B-Algorithm
polynomials	I-Algorithm
of	O
degree	O
six	O
in	O
normalized	O
form	O
can	O
be	O
found	O
in	O
.	O
</s>
<s>
Beyond	O
the	O
above	O
examples	O
,	O
the	O
following	O
list	O
,	O
while	O
not	O
exhaustive	O
,	O
contains	O
almost	O
all	O
of	O
the	O
known	O
major	O
classes	O
of	O
permutation	B-Algorithm
polynomials	I-Algorithm
over	O
finite	O
fields	O
.	O
</s>
<s>
If	O
is	O
an	O
extension	O
of	O
of	O
degree	O
,	O
then	O
the	O
linearized	O
polynomial	O
with	O
in	O
,	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
on	O
over	O
.	O
</s>
<s>
The	O
linearized	O
polynomials	O
that	O
are	O
permutation	B-Algorithm
polynomials	I-Algorithm
over	O
form	O
a	O
group	O
under	O
the	O
operation	O
of	O
composition	O
modulo	O
,	O
which	O
is	O
known	O
as	O
the	O
Betti-Mathieu	O
group	O
,	O
isomorphic	O
to	O
the	O
general	O
linear	O
group	O
.	O
</s>
<s>
Only	O
a	O
few	O
other	O
specific	O
classes	O
of	O
permutation	B-Algorithm
polynomials	I-Algorithm
over	O
have	O
been	O
characterized	O
.	O
</s>
<s>
An	O
exceptional	O
polynomial	O
over	O
is	O
a	O
polynomial	O
in	O
which	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
on	O
for	O
infinitely	O
many	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
polynomial	I-Algorithm
over	O
of	O
degree	O
at	O
most	O
is	O
exceptional	O
over	O
.	O
</s>
<s>
Every	O
permutation	B-Algorithm
of	O
is	O
induced	O
by	O
an	O
exceptional	O
polynomial	O
.	O
</s>
<s>
If	O
a	O
polynomial	O
with	O
integer	O
coefficients	O
(	O
i.e.	O
,	O
in	O
)	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
over	O
for	O
infinitely	O
many	O
primes	O
,	O
then	O
it	O
is	O
the	O
composition	O
of	O
linear	O
and	O
Dickson	O
polynomials	O
.	O
</s>
<s>
In	O
finite	O
geometry	O
coordinate	O
descriptions	O
of	O
certain	O
point	O
sets	O
can	O
provide	O
examples	O
of	O
permutation	B-Algorithm
polynomials	I-Algorithm
of	O
higher	O
degree	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
points	O
forming	O
an	O
oval	O
in	O
a	O
finite	O
projective	O
plane	O
,	O
with	O
a	O
power	O
of	O
2	O
,	O
can	O
be	O
coordinatized	O
in	O
such	O
a	O
way	O
that	O
the	O
relationship	O
between	O
the	O
coordinates	O
is	O
given	O
by	O
an	O
o-polynomial	O
,	O
which	O
is	O
a	O
special	O
type	O
of	O
permutation	B-Algorithm
polynomial	I-Algorithm
over	O
the	O
finite	O
field	O
.	O
</s>
<s>
The	O
problem	O
of	O
testing	O
whether	O
a	O
given	O
polynomial	O
over	O
a	O
finite	O
field	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
can	O
be	O
solved	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
A	O
polynomial	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
in	O
variables	O
over	O
if	O
the	O
equation	O
has	O
exactly	O
solutions	O
in	O
for	O
each	O
.	O
</s>
<s>
For	O
the	O
finite	O
ring	O
Z/nZ	O
one	O
can	O
construct	O
quadratic	O
permutation	B-Algorithm
polynomials	I-Algorithm
.	O
</s>
<s>
Actually	O
it	O
is	O
possible	O
if	O
and	O
only	O
if	O
n	O
is	O
divisible	O
by	O
p2	O
for	O
some	O
prime	O
number	O
p	O
.	O
The	O
construction	O
is	O
surprisingly	O
simple	O
,	O
nevertheless	O
it	O
can	O
produce	O
permutations	B-Algorithm
with	O
certain	O
good	O
properties	O
.	O
</s>
<s>
That	O
is	O
why	O
it	O
has	O
been	O
used	O
in	O
the	O
interleaver	O
component	O
of	O
turbo	B-Error_Name
codes	I-Error_Name
in	O
3GPP	O
Long	O
Term	O
Evolution	O
mobile	O
telecommunication	O
standard	O
(	O
see	O
3GPP	O
technical	O
specification	O
36.212	O
e.g.	O
</s>
<s>
Z/pZ	O
)	O
such	O
polynomial	O
defines	O
a	O
permutation	B-Algorithm
only	O
in	O
the	O
case	O
a	O
=	O
0	O
and	O
b	O
not	O
equal	O
to	O
zero	O
.	O
</s>
<s>
Lemma	O
:	O
for	O
k>1	O
,	O
p>2	O
(	O
Z/pkZ	O
)	O
such	O
polynomial	O
defines	O
a	O
permutation	B-Algorithm
if	O
and	O
only	O
if	O
and	O
.	O
</s>
<s>
Lemma	O
:	O
any	O
polynomial	O
defines	O
a	O
permutation	B-Algorithm
for	O
the	O
ring	O
Z/nZ	O
if	O
and	O
only	O
if	O
all	O
the	O
polynomials	O
defines	O
the	O
permutations	B-Algorithm
for	O
all	O
rings	O
,	O
where	O
are	O
remainders	O
of	O
modulo	O
.	O
</s>
<s>
As	O
a	O
corollary	O
one	O
can	O
construct	O
plenty	O
quadratic	O
permutation	B-Algorithm
polynomials	I-Algorithm
using	O
the	O
following	O
simple	O
construction	O
.	O
</s>
<s>
Then	O
such	O
polynomial	O
defines	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
To	O
see	O
this	O
we	O
observe	O
that	O
for	O
all	O
primes	O
pi	O
,	O
i	O
>	O
1	O
,	O
the	O
reduction	O
of	O
this	O
quadratic	O
polynomial	O
modulo	O
pi	O
is	O
actually	O
linear	O
polynomial	O
and	O
hence	O
is	O
permutation	B-Algorithm
by	O
trivial	O
reason	O
.	O
</s>
<s>
For	O
the	O
first	O
prime	O
number	O
we	O
should	O
use	O
the	O
lemma	O
discussed	O
previously	O
to	O
see	O
that	O
it	O
defines	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
A	O
polynomial	O
g(x )	O
for	O
the	O
ring	O
Z/pkZ	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
if	O
and	O
only	O
if	O
it	O
permutes	O
the	O
finite	O
field	O
Z/pZ	O
and	O
for	O
all	O
x	O
in	O
Z/pkZ	O
,	O
where	O
g(x )	O
is	O
the	O
formal	O
derivative	O
of	O
g(x )	O
.	O
</s>
<s>
The	O
term	O
"	O
Schur	O
's	O
conjecture	O
"	O
refers	O
to	O
the	O
assertion	O
that	O
,	O
if	O
a	O
polynomial	O
f	O
defined	O
over	O
K	O
is	O
a	O
permutation	B-Algorithm
polynomial	I-Algorithm
on	O
R/P	O
for	O
infinitely	O
many	O
prime	O
ideals	O
P	O
,	O
then	O
f	O
is	O
the	O
composition	O
of	O
Dickson	O
polynomials	O
,	O
degree-one	O
polynomials	O
,	O
and	O
polynomials	O
of	O
the	O
form	O
xk	O
.	O
</s>
