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Permutation Polynomials over Finite Fields
Z¨
ulf¨
ukar Saygı
Department of Mathematics,
TOBB University of Economics and Technology,
Ankara, Turkey.
10 April 2015
Z¨
ulf¨
ukar Saygı
Permutation Polynomials over F.F.
Outline
Basic Definitions and Notations
Some Known Results
Motivation
Some New Results and Open Problems
Z¨
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Permutation Polynomials over F.F.
Notations
q be a positive power of a prime,
Fq be a finite field with q elements,
F∗q = Fq \ {0},
Fq [x] be the polynomial ring with the variable x,
Sn be the symmetric group of order n,
Tr be the trace map from Fqk to Fq ,
k−1
where Tr(x) = x + x q + · · · + x q .
Z¨
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Permutation Polynomials over F.F.
Permutation Polynomials
A polynomial f ∈ Fq [x] is called a permutation polynomial
(PP) of Fq if x → f (x) is a permutation of Fq .
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Permutation Polynomials
A polynomial f ∈ Fq [x] is called a permutation polynomial
(PP) of Fq if x → f (x) is a permutation of Fq .
A PP correspond to an element of the symmetric group Sq .
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Permutation Polynomials
A polynomial f ∈ Fq [x] is called a permutation polynomial
(PP) of Fq if x → f (x) is a permutation of Fq .
A PP correspond to an element of the symmetric group Sq .
There are q! PPs of Fq , all of which are given by the Lagrange
interpolation
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Permutation Polynomials
A polynomial f ∈ Fq [x] is called a permutation polynomial
(PP) of Fq if x → f (x) is a permutation of Fq .
A PP correspond to an element of the symmetric group Sq .
There are q! PPs of Fq , all of which are given by the Lagrange
interpolation
Given a permutation g of Fq , the unique permutation
polynomial Pg (x) of Fq of degree at most q − 1:
X
Pg (x) =
g (a) 1 − (x − a)q−1
a∈Fq
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Permutation Polynomials over F.F.
A Remark
If f is a PP and a 6= 0, b 6= 0, c ∈ Fq , then f1 = af (bx + c) is
also a PP.
Z¨
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Permutation Polynomials over F.F.
A Remark
If f is a PP and a 6= 0, b 6= 0, c ∈ Fq , then f1 = af (bx + c) is
also a PP.
By suitably choosing a, b, c we can arrange to have f1 in
normalized form
f1 is monic,
f1 (0) = 0,
when the degree n of f1 is not divisible by char (Fq ), the
coefficient of x n−1 is 0.
Z¨
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Permutation Polynomials over F.F.
Some well known classes of PPs
Every linear polynomial over Fq is a PP.
Monomials: x n is a PP of Fq iff gcd(n, q − 1) = 1.
Z¨
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Permutation Polynomials over F.F.
Some well known classes of PPs
Every linear polynomial over Fq is a PP.
Monomials: x n is a PP of Fq iff gcd(n, q − 1) = 1.
Dickson: For a ∈ F∗q , the polynomial
bn/2c
Dn (x, a) =
X
i=0
n
n−i
(−a)i x n−2i
n−i
i
is a PP of Fq iff (n, q 2 − 1) = 1.
Z¨
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Permutation Polynomials over F.F.
Some well known classes of PPs
Every linear polynomial over Fq is a PP.
Monomials: x n is a PP of Fq iff gcd(n, q − 1) = 1.
Dickson: For a ∈ F∗q , the polynomial
bn/2c
Dn (x, a) =
X
i=0
n
n−i
(−a)i x n−2i
n−i
i
is a PP of Fq iff (n, q 2 − 1) = 1.
Pn−1
q s ∈ F n [x] is a
Linearized: The polynomial
L(x)
=
q
s=0 as x
j q
n
PP of Fq iff det ai−j 6= 0, 0 ≤ i, j ≤ n − 1.
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Permutation Polynomials over F.F.
More Examples
For odd q, f (x) = x (q+1)/2 + ax is a PP of Fq iff a2 − 1 is a
nonzero square in Fq .
f (x) = x r (g (x d ))(q−1)/d is a PP of Fq if gcd(r , q − 1) = 1,
d | q − 1, and g (x d ) has no nonzero root in Fq .
Z¨
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Permutation Polynomials over F.F.
More Examples
For odd q, f (x) = x (q+1)/2 + ax is a PP of Fq iff a2 − 1 is a
nonzero square in Fq .
f (x) = x r (g (x d ))(q−1)/d is a PP of Fq if gcd(r , q − 1) = 1,
d | q − 1, and g (x d ) has no nonzero root in Fq .
Note that if f (x) and g (x) are PPs of Fq then f (g (x)) is a
PP of Fq .
Z¨
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Permutation Polynomials over F.F.
Criteria for the PPs
For f ∈ Fq [x], the following statements are equivalent:
f is a PP of Fq
For each y ∈ Fq , f (x) = y has at least one solution x ∈ Fq
For each y ∈ Fq , f (x) = y has at most one solution x ∈ Fq
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Criteria for the PPs
For f ∈ Fq [x], the following statements are equivalent:
f is a PP of Fq
For each y ∈ Fq , f (x) = y has at least one solution x ∈ Fq
For each y ∈ Fq , f (x) = y has at most one solution x ∈ Fq
For all nontrivial additive characters χ of Fq we have
X
χ(f (a)) = 0
a∈Fq
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Criteria for the PPs
For f ∈ Fq [x], the following statements are equivalent:
f is a PP of Fq
For each y ∈ Fq , f (x) = y has at least one solution x ∈ Fq
For each y ∈ Fq , f (x) = y has at most one solution x ∈ Fq
For all nontrivial additive characters χ of Fq we have
X
χ(f (a)) = 0
a∈Fq
Note that
P
a∈Fq
χ(f (a)) =
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P
a∈Fq
χ(a) = 0
Permutation Polynomials over F.F.
Hermite’s criterion
f is a PP of Fq iff
X
x∈Fq
s
f (x) =
0
−1
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if 0 ≤ s ≤ q − 2,
if s = q − 1.
Permutation Polynomials over F.F.
Hermite’s criterion
f is a PP of Fq iff
X
s
f (x) =
x∈Fq
0
−1
if 0 ≤ s ≤ q − 2,
if s = q − 1.
f is a PP of Fq iff
1
2
f (x) has exactly one root in Fq ,
For each integer t with 1 ≤ t ≤ q − 2 and p 6 | t, the reduction
t
of (f (x)) mod (x q − x) has degree at most q − 2
where p = char (Fq ).
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Hermite’s criterion
f is a PP of Fq iff
X
s
f (x) =
x∈Fq
0
−1
if 0 ≤ s ≤ q − 2,
if s = q − 1.
f is a PP of Fq iff
1
2
f (x) has exactly one root in Fq ,
For each integer t with 1 ≤ t ≤ q − 2 and p 6 | t, the reduction
t
of (f (x)) mod (x q − x) has degree at most q − 2
where p = char (Fq ).
Difficult to apply for a general polynomial.
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Enumeration of PPs
Hermite’s criterion was used by Dickson to obtain all
normalized PPs of degree at most 5.
A list of PPs of degree 6 over finite fields with odd
characteristic can be found in [D].
[D] L. E. Dickson, The analytic representation of substitutions
on a power of a prime number of letters with a discussion of
the linear group, Ann. of Math. 11 (1896/97), 65–120.
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Enumeration of PPs
Hermite’s criterion was used by Dickson to obtain all
normalized PPs of degree at most 5.
A list of PPs of degree 6 over finite fields with odd
characteristic can be found in [D].
[D] L. E. Dickson, The analytic representation of substitutions
on a power of a prime number of letters with a discussion of
the linear group, Ann. of Math. 11 (1896/97), 65–120.
A list of PPs of degree 6 and 7 over finite fields with char=2
is presented in [LCX].
[LCX] J. Li, D. B. Chandler, and Q. Xiang, Permutation
polynomials of degree 6 or 7 over finite fields of characteristic
2, Finite Fields Appl. 16 (2010) 406–419.
Z¨
ulf¨
ukar Saygı
Permutation Polynomials over F.F.
Enumeration of PPs
Hermite’s criterion was used by Dickson to obtain all
normalized PPs of degree at most 5.
A list of PPs of degree 6 over finite fields with odd
characteristic can be found in [D].
[D] L. E. Dickson, The analytic representation of substitutions
on a power of a prime number of letters with a discussion of
the linear group, Ann. of Math. 11 (1896/97), 65–120.
A list of PPs of degree 6 and 7 over finite fields with char=2
is presented in [LCX].
[LCX] J. Li, D. B. Chandler, and Q. Xiang, Permutation
polynomials of degree 6 or 7 over finite fields of characteristic
2, Finite Fields Appl. 16 (2010) 406–419.
All monic PPs of degree 6 in the normalized form is presented
in [SW].
[SW] C. J. Shallue and I. M. Wanless, Permutation
polynomials and orthomorphism polynomials of degree six,
Finite Fields and Their Applications 20 (2013) 84–92
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ukar Saygı
Permutation Polynomials over F.F.
An Open Problem
Let Nn (q) denote the number of PPs of Fq which have degree
n.
Trivial boundary conditions:
N1 (q) = q(q − 1),
Nn (q) = 0 if n 6= 1 is a divisor of q − 1,
Pq−1
n=1 Nn (q) = q!.
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
An Open Problem
Let Nn (q) denote the number of PPs of Fq which have degree
n.
Trivial boundary conditions:
N1 (q) = q(q − 1),
Nn (q) = 0 if n 6= 1 is a divisor of q − 1,
Pq−1
n=1 Nn (q) = q!.
Problem: Find Nn (q).
R. Lidl and G. L. Mullen, When does a polynomial over a finite
field permute the elements of the field?, II, Amer. Math. Monthly
100 (1993) 71–74.
Z¨
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Permutation Polynomials over F.F.
Motivation
m a positive integer,
F2 finite field of order 2.
m
f : Fm
2 → F2
(
u
Nf (u, v ) := #
v
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= x + y;
= f (x) + f (y ).
Permutation Polynomials over F.F.
Motivation
m a positive integer,
F2 finite field of order 2.
m
f : Fm
2 → F2
(
u
Nf (u, v ) := #
v
= x + y;
= f (x) + f (y ).
u 6= 0 =⇒ Nf (u, v ) = 0 or 2
→ f is almost perfect non-linear [APN] .
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Motivation
m a positive integer,
F2 finite field of order 2.
m
f : Fm
2 → F2
(
u
Nf (u, v ) := #
v
= x + y;
= f (x) + f (y ).
u 6= 0 =⇒ Nf (u, v ) = 0 or 2
→ f is almost perfect non-linear [APN] .
(Affine Equivalence) Let A, B, C be three affine
transformations of Fm
2 . If A, B are permutations then
f is APN ⇐⇒ A ◦ f ◦ B + C is APN
Z¨
ulf¨
ukar Saygı
Permutation Polynomials over F.F.
Motivation
Flat characterization of APNs
(
x +y +z +t =0
=⇒ f (x) + f (y ) + f (z) + f (t) 6= 0
all distinct
then f is [APN] .
Z¨
ulf¨
ukar Saygı
Permutation Polynomials over F.F.
Motivation
Flat characterization of APNs
(
x +y +z +t =0
=⇒ f (x) + f (y ) + f (z) + f (t) 6= 0
all distinct
then f is [APN] .
Code Characterization


1
...
1
...
1
...
x
...
1 
Hf =  0
f (0) . . . f (x) . . . f (1)
if the minimal distance of code(f ) > 4 then f is [APN] .
Z¨
ulf¨
ukar Saygı
Permutation Polynomials over F.F.
Motivation
Flat characterization of APNs
(
x +y +z +t =0
=⇒ f (x) + f (y ) + f (z) + f (t) 6= 0
all distinct
then f is [APN] .
Code Characterization


1
...
1
...
1
...
x
...
1 
Hf =  0
f (0) . . . f (x) . . . f (1)
if the minimal distance of code(f ) > 4 then f is [APN] .
(The code is double-error-correcting (no fewer than 5 cols sum to 0).)
Z¨
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Permutation Polynomials over F.F.
Motivation
Dobbertin constructed several classes of PPs over finite fields
of even characteristic and used them to prove several
conjectures on APN monomials.
H. Dobbertin, Almost perfect nonlinear power functions on
GF (2n ): the Niho case, Inform. and Comput. 151 (1999)
57–72.
H. Dobbertin, Almost perfect nonlinear power functions on
GF (2n ): the Welch case, IEEE Trans. Inform. Theory 45
(1999) 1271–1275.
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Permutation Polynomials over F.F.
Motivation
The existence of APN permutations on F22n is a long-term
open problem.
Hou proved that there are no APN permutations over F24 and
there are no APN permutations on F22n with coefficients in
F2n .
X.-D. Hou, Affinity of permutations of Fn2 , Discrete Appl.
Math. 154 (2006) 313–325.
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Motivation
The existence of APN permutations on F22n is a long-term
open problem.
Hou proved that there are no APN permutations over F24 and
there are no APN permutations on F22n with coefficients in
F2n .
X.-D. Hou, Affinity of permutations of Fn2 , Discrete Appl.
Math. 154 (2006) 313–325.
Recently, Dillon presented the first APN permutation over F26 .
K. A. Browning, J. F. Dillon, M. T. McQuistan, and A. J.
Wolfe, An APN permutation in dimension six, In Finite Fields:
Theory and Applications, volume 518 of Contemp. Math.,
33–42, Amer. Math. Soc., Providence, RI, 2010.
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Permutation Polynomials over F.F.
Motivation
The existence of APN permutations on F22n is a long-term
open problem.
Hou proved that there are no APN permutations over F24 and
there are no APN permutations on F22n with coefficients in
F2n .
X.-D. Hou, Affinity of permutations of Fn2 , Discrete Appl.
Math. 154 (2006) 313–325.
Recently, Dillon presented the first APN permutation over F26 .
K. A. Browning, J. F. Dillon, M. T. McQuistan, and A. J.
Wolfe, An APN permutation in dimension six, In Finite Fields:
Theory and Applications, volume 518 of Contemp. Math.,
33–42, Amer. Math. Soc., Providence, RI, 2010.
Open Problem Is there any APN permutation on F22n for
n ≥ 4.
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Motivation
The Kloosterman sum K (a) over F2n is defined for any
a ∈ F2n by
X
1
(−1)Tr(ax+ x )
K (a) =
a∈F∗2n
Z¨
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Permutation Polynomials over F.F.
Motivation
The Kloosterman sum K (a) over F2n is defined for any
a ∈ F2n by
X
1
(−1)Tr(ax+ x )
K (a) =
a∈F∗2n
Shin, Kumar and Helleseth found that the existence of certain
3-designs in the Goethals code of length 2n , n odd, over Z4
was equivalent to the identity
3 a
a
=
K
∀a ∈ F2n \ {1}
K
4
1+a
1 + a4
and they proved this identity for all odd values of n.
This relation was extended to the case n even by Helleseth and
Zinoviev.
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Permutation Polynomials over F.F.
Special PPs
Helleseth and Zinoviev used the PPs
2l
1
+ x over F2n
x2 + x + δ
to derive new identities of Kloosterman sums over F2n ,
where δ ∈ F2n with Tr(δ) = 1 and l ∈ {0, 1}.
Z¨
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ukar Saygı
Permutation Polynomials over F.F.
Special PPs
Helleseth and Zinoviev used the PPs
2l
1
+ x over F2n
x2 + x + δ
to derive new identities of Kloosterman sums over F2n ,
where δ ∈ F2n with Tr(δ) = 1 and l ∈ {0, 1}.
Recently, PPs with the form
s
i
f (x) = x p − x + δ + L(x)
over the finite field Fq have been extensively studied
where , i, s ∈ Z+ , δ ∈ Fq , char (Fq ) = p and L(x) is a
linearized polynomial in Fq [x].
Z¨
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Permutation Polynomials over F.F.
Some PPs of the form
i
xp − x + δ
s
+ L(x)
T. Helleseth, V. Zinoviev, New Kloosterman sums identities
over F2m for all m, Finite Fields Appl. 9 (2003) 187-193.
J. Yuan, C. Ding, Four classes of permutation polynomials of
F2m , Finite Fields Appl. 13 (2007) 869-876.
J. Yuan, C. Ding, H. Wang, J. Pieprzyk, Permutation
polynomials of the form (x p − x + δ)s + L(x), Finite Fields
Appl. 14 (2008) 482-493.
X. Zeng, X. Zhu, L. Hu, Two new permutation polynomials
k
with the form (x 2 + x + δ)s + x over F2n , Appl. Algebra Eng.
Commun. Comput. 21 (2010) 145-150.
N. Li, T. Helleseth, X. Tang, Further results on a class of
permutation polynomials over finite fields, Finite Fields Appl.
22 (2013) 16-23.
Z. Tu, X. Zeng, C.Li, T. Helleseth, Permutation polynomials
m
of the form (x p − x + δ)s + L(x) over the finite field Fp2m of
odd characteristic, Finite Fields Appl. 34 (2015) 20-35.
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Permutation Polynomials over F.F.
Known Explicit PPs
Let p be an odd prime
1
For positive integers n and k and δ ∈ Fpn ,
pn2+1
k
k
p
x −x +δ
+ x p + x is a PP of Fpn .
2
For positive integer k and δ ∈ F33k with Tr33k /3k (δ) = 0,
33k2−1 +3k
k
k
x3 − x + δ
+ x 3 + x is a PP of F33k .
3
For positive integers n and k with n|4k and δ ∈ Fpn ,
k
pn2−1 +p2k
k
p
x −x +δ
± (x p + x) is a PP of Fpn .
For a positive integer m and for any δ ∈ F32m ,
2·3m −1
m
m
x3 − x + δ
+ x 3 + x is a PP of F32m .
For a positive integer m and δ ∈ F32m , if (Tr32m /3m (δ))2 + 1 = 0
3m +2
m
or a square in F3m , x 3 − x + δ
+ x is a PP of F32m .
4
5
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Permutation Polynomials over F.F.
New PPs
Theorem
Let n = (t − 1)k, where k is a positive integer, t is an odd integer,
gcd(3, t) = 1 and δ ∈ F∗3n .
f (x) = (x 3
( t−1
2 )k
( t−1
2 )k
− x + δ)s + x and
( t−1 )k
− x + δ)s + x 3 2 + x
g (x) = (x 3
are permutation polynomials over F3n with s =
Tr (δ) = 0.
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3n −1
t
+ 1 and
Permutation Polynomials over F.F.
New PPs
Theorem
Let n = 4k, where k is a positive integer and δ ∈ F∗3n .
2k
f (x) = (x 3 − x + δ)s + x is a permutation polynomial over F3n
for the following cases:
Let k be a positive integer,jw be the
k generator of F3n ,
3n/2 −1
3n −1
s = 3( 5 ) + 1 and ` = 2
+ 3n/4 .
5
Then for δ = w `
over F3n .
(mod 2`)
∈ F3n and Tr (δ) = 0, f (x) is a PP
n
Let k = 1 and s = ( 3 5−1 ) + 1. For each δ ∈ F∗3n with
Tr (δ) = 0, then f (x) is a PP over F3n .
n
Let k = 1 and s = 2( 3 5−1 ) + 1. Then for each δ ∈ F∗3n f (x) is
a PP over F3n .
k
For this case f (x) + x and f (x) + x 3 are also a permutation
polynomial over F3n .
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Permutation Polynomials over F.F.
New PPs
Theorem
Let n = 4k, where k is a positive integer and δ ∈ F∗7n .
n
Let w be the
of F7n , s = i( 7 5−1 ) + 1, where i ∈ {1, 2, 3}
j generator
k
and ` = 2
7n/2 −1
+ 7n/4 .
5
= w ` (mod 2`) ∈
Then for δ
F7n and Tr (δ) = 0,
2k
7
s
f (x) = (x − x + δ) + x is a permutation polynomial over F7n .
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Permutation Polynomials over F.F.
THANKS...
Z¨
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ukar Saygı
Permutation Polynomials over F.F.