Permutation Polynomials over Finite Fields
Zulfukar Saygı
Department of Mathematics,TOBB University of Economics and Technology,
Ankara, Turkey.
10 April 2015
Zulfukar Saygı Permutation Polynomials over F.F.
Outline
Basic Definitions and Notations
Some Known Results
Motivation
Some New Results and Open Problems
Zulfukar Saygı 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,
where Tr(x) = x + xq + · · ·+ xqk−1.
Zulfukar 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 Lagrangeinterpolation
Given a permutation g of Fq, the unique permutationpolynomial Pg (x) of Fq of degree at most q − 1:
Pg (x) =∑a∈Fq
g(a)(
1 − (x − a)q−1)
Zulfukar 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 Lagrangeinterpolation
Given a permutation g of Fq, the unique permutationpolynomial Pg (x) of Fq of degree at most q − 1:
Pg (x) =∑a∈Fq
g(a)(
1 − (x − a)q−1)
Zulfukar 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 Lagrangeinterpolation
Given a permutation g of Fq, the unique permutationpolynomial Pg (x) of Fq of degree at most q − 1:
Pg (x) =∑a∈Fq
g(a)(
1 − (x − a)q−1)
Zulfukar 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 Lagrangeinterpolation
Given a permutation g of Fq, the unique permutationpolynomial Pg (x) of Fq of degree at most q − 1:
Pg (x) =∑a∈Fq
g(a)(
1 − (x − a)q−1)
Zulfukar Saygı 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) isalso a PP.
By suitably choosing a, b, c we can arrange to have f1 innormalized form
f1 is monic,f1(0) = 0,when the degree n of f1 is not divisible by char(Fq), thecoefficient of xn−1 is 0.
Zulfukar Saygı 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) isalso a PP.
By suitably choosing a, b, c we can arrange to have f1 innormalized form
f1 is monic,f1(0) = 0,when the degree n of f1 is not divisible by char(Fq), thecoefficient of xn−1 is 0.
Zulfukar Saygı Permutation Polynomials over F.F.
Some well known classes of PPs
Every linear polynomial over Fq is a PP.
Monomials: xn is a PP of Fq iff gcd(n, q − 1) = 1.
Dickson: For a ∈ F∗q, the polynomial
Dn(x , a) =
bn/2c∑i=0
n
n − i
(n − i
i
)(−a)ixn−2i
is a PP of Fq iff (n, q2 − 1) = 1.
Linearized: The polynomial L(x) =∑n−1
s=0 asxqs ∈ Fqn [x ] is a
PP of Fqn iff det(aq
j
i−j
)6= 0, 0 ≤ i , j ≤ n − 1.
Zulfukar Saygı Permutation Polynomials over F.F.
Some well known classes of PPs
Every linear polynomial over Fq is a PP.
Monomials: xn is a PP of Fq iff gcd(n, q − 1) = 1.
Dickson: For a ∈ F∗q, the polynomial
Dn(x , a) =
bn/2c∑i=0
n
n − i
(n − i
i
)(−a)ixn−2i
is a PP of Fq iff (n, q2 − 1) = 1.
Linearized: The polynomial L(x) =∑n−1
s=0 asxqs ∈ Fqn [x ] is a
PP of Fqn iff det(aq
j
i−j
)6= 0, 0 ≤ i , j ≤ n − 1.
Zulfukar Saygı Permutation Polynomials over F.F.
Some well known classes of PPs
Every linear polynomial over Fq is a PP.
Monomials: xn is a PP of Fq iff gcd(n, q − 1) = 1.
Dickson: For a ∈ F∗q, the polynomial
Dn(x , a) =
bn/2c∑i=0
n
n − i
(n − i
i
)(−a)ixn−2i
is a PP of Fq iff (n, q2 − 1) = 1.
Linearized: The polynomial L(x) =∑n−1
s=0 asxqs ∈ Fqn [x ] is a
PP of Fqn iff det(aq
j
i−j
)6= 0, 0 ≤ i , j ≤ n − 1.
Zulfukar Saygı 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 anonzero square in Fq.
f (x) = x r (g(xd))(q−1)/d is a PP of Fq if gcd(r , q − 1) = 1,d | q − 1, and g(xd) has no nonzero root in Fq.
Note that if f (x) and g(x) are PPs of Fq then f (g(x)) is aPP of Fq.
Zulfukar Saygı 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 anonzero square in Fq.
f (x) = x r (g(xd))(q−1)/d is a PP of Fq if gcd(r , q − 1) = 1,d | q − 1, and g(xd) has no nonzero root in Fq.
Note that if f (x) and g(x) are PPs of Fq then f (g(x)) is aPP of Fq.
Zulfukar 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∑a∈Fq
χ(f (a)) = 0
Note that∑
a∈Fqχ(f (a)) =
∑a∈Fq
χ(a) = 0
Zulfukar 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∑a∈Fq
χ(f (a)) = 0
Note that∑
a∈Fqχ(f (a)) =
∑a∈Fq
χ(a) = 0
Zulfukar 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∑a∈Fq
χ(f (a)) = 0
Note that∑
a∈Fqχ(f (a)) =
∑a∈Fq
χ(a) = 0
Zulfukar Saygı Permutation Polynomials over F.F.
Hermite’s criterion
f is a PP of Fq iff
∑x∈Fq
f (x)s =
{0 if 0 ≤ s ≤ q − 2,−1 if s = q − 1.
f is a PP of Fq iff1 f (x) has exactly one root in Fq,2 For each integer t with 1 ≤ t ≤ q − 2 and p 6 | t, the reduction
of (f (x))t mod (xq − x) has degree at most q − 2
where p = char(Fq).
Difficult to apply for a general polynomial.
Zulfukar Saygı Permutation Polynomials over F.F.
Hermite’s criterion
f is a PP of Fq iff
∑x∈Fq
f (x)s =
{0 if 0 ≤ s ≤ q − 2,−1 if s = q − 1.
f is a PP of Fq iff1 f (x) has exactly one root in Fq,2 For each integer t with 1 ≤ t ≤ q − 2 and p 6 | t, the reduction
of (f (x))t mod (xq − x) has degree at most q − 2
where p = char(Fq).
Difficult to apply for a general polynomial.
Zulfukar Saygı Permutation Polynomials over F.F.
Hermite’s criterion
f is a PP of Fq iff
∑x∈Fq
f (x)s =
{0 if 0 ≤ s ≤ q − 2,−1 if s = q − 1.
f is a PP of Fq iff1 f (x) has exactly one root in Fq,2 For each integer t with 1 ≤ t ≤ q − 2 and p 6 | t, the reduction
of (f (x))t mod (xq − x) has degree at most q − 2
where p = char(Fq).
Difficult to apply for a general polynomial.
Zulfukar Saygı Permutation Polynomials over F.F.
Enumeration of PPs
Hermite’s criterion was used by Dickson to obtain allnormalized PPs of degree at most 5.
A list of PPs of degree 6 over finite fields with oddcharacteristic can be found in [D].
[D] L. E. Dickson, The analytic representation of substitutionson a power of a prime number of letters with a discussion ofthe linear group, Ann. of Math. 11 (1896/97), 65–120.
A list of PPs of degree 6 and 7 over finite fields with char=2is presented in [LCX].
[LCX] J. Li, D. B. Chandler, and Q. Xiang, Permutationpolynomials of degree 6 or 7 over finite fields of characteristic2, Finite Fields Appl. 16 (2010) 406–419.
All monic PPs of degree 6 in the normalized form is presentedin [SW].
[SW] C. J. Shallue and I. M. Wanless, Permutationpolynomials and orthomorphism polynomials of degree six,Finite Fields and Their Applications 20 (2013) 84–92
Zulfukar Saygı Permutation Polynomials over F.F.
Enumeration of PPs
Hermite’s criterion was used by Dickson to obtain allnormalized PPs of degree at most 5.
A list of PPs of degree 6 over finite fields with oddcharacteristic can be found in [D].
[D] L. E. Dickson, The analytic representation of substitutionson a power of a prime number of letters with a discussion ofthe linear group, Ann. of Math. 11 (1896/97), 65–120.
A list of PPs of degree 6 and 7 over finite fields with char=2is presented in [LCX].
[LCX] J. Li, D. B. Chandler, and Q. Xiang, Permutationpolynomials of degree 6 or 7 over finite fields of characteristic2, Finite Fields Appl. 16 (2010) 406–419.
All monic PPs of degree 6 in the normalized form is presentedin [SW].
[SW] C. J. Shallue and I. M. Wanless, Permutationpolynomials and orthomorphism polynomials of degree six,Finite Fields and Their Applications 20 (2013) 84–92
Zulfukar Saygı Permutation Polynomials over F.F.
Enumeration of PPs
Hermite’s criterion was used by Dickson to obtain allnormalized PPs of degree at most 5.
A list of PPs of degree 6 over finite fields with oddcharacteristic can be found in [D].
[D] L. E. Dickson, The analytic representation of substitutionson a power of a prime number of letters with a discussion ofthe linear group, Ann. of Math. 11 (1896/97), 65–120.
A list of PPs of degree 6 and 7 over finite fields with char=2is presented in [LCX].
[LCX] J. Li, D. B. Chandler, and Q. Xiang, Permutationpolynomials of degree 6 or 7 over finite fields of characteristic2, Finite Fields Appl. 16 (2010) 406–419.
All monic PPs of degree 6 in the normalized form is presentedin [SW].
[SW] C. J. Shallue and I. M. Wanless, Permutationpolynomials and orthomorphism polynomials of degree six,Finite Fields and Their Applications 20 (2013) 84–92
Zulfukar Saygı Permutation Polynomials over F.F.
An Open Problem
Let Nn(q) denote the number of PPs of Fq which have degreen.
Trivial boundary conditions:
N1(q) = q(q − 1),Nn(q) = 0 if n 6= 1 is a divisor of q − 1,∑q−1
n=1 Nn(q) = q!.
Problem: Find Nn(q).R. Lidl and G. L. Mullen, When does a polynomial over a finitefield permute the elements of the field?, II, Amer. Math. Monthly100 (1993) 71–74.
Zulfukar Saygı Permutation Polynomials over F.F.
An Open Problem
Let Nn(q) denote the number of PPs of Fq which have degreen.
Trivial boundary conditions:
N1(q) = q(q − 1),Nn(q) = 0 if n 6= 1 is a divisor of q − 1,∑q−1
n=1 Nn(q) = q!.
Problem: Find Nn(q).R. Lidl and G. L. Mullen, When does a polynomial over a finitefield permute the elements of the field?, II, Amer. Math. Monthly100 (1993) 71–74.
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
m a positive integer,
F2 finite field of order 2.
f : Fm2 → Fm
2
Nf (u, v) := #
{u = x + y ;
v = 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 affinetransformations of Fm
2 . If A, B are permutations then
f is APN ⇐⇒ A ◦ f ◦ B + C is APN
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
m a positive integer,
F2 finite field of order 2.
f : Fm2 → Fm
2
Nf (u, v) := #
{u = x + y ;
v = 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 affinetransformations of Fm
2 . If A, B are permutations then
f is APN ⇐⇒ A ◦ f ◦ B + C is APN
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
m a positive integer,
F2 finite field of order 2.
f : Fm2 → Fm
2
Nf (u, v) := #
{u = x + y ;
v = 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 affinetransformations of Fm
2 . If A, B are permutations then
f is APN ⇐⇒ A ◦ f ◦ B + C is APN
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
Flat characterization of APNs{x + y + z + t = 0
all distinct=⇒ f (x) + f (y) + f (z) + f (t) 6= 0
then f is [APN] .
Code Characterization
Hf =
1 . . . 1 . . . 10 . . . x . . . 1
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).)
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
Flat characterization of APNs{x + y + z + t = 0
all distinct=⇒ f (x) + f (y) + f (z) + f (t) 6= 0
then f is [APN] .
Code Characterization
Hf =
1 . . . 1 . . . 10 . . . x . . . 1
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).)
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
Flat characterization of APNs{x + y + z + t = 0
all distinct=⇒ f (x) + f (y) + f (z) + f (t) 6= 0
then f is [APN] .
Code Characterization
Hf =
1 . . . 1 . . . 10 . . . x . . . 1
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).)
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
Dobbertin constructed several classes of PPs over finite fieldsof even characteristic and used them to prove severalconjectures on APN monomials.
H. Dobbertin, Almost perfect nonlinear power functions onGF (2n): the Niho case, Inform. and Comput. 151 (1999)57–72.H. Dobbertin, Almost perfect nonlinear power functions onGF (2n): the Welch case, IEEE Trans. Inform. Theory 45(1999) 1271–1275.
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
The existence of APN permutations on F22n is a long-termopen problem.
Hou proved that there are no APN permutations over F24 andthere are no APN permutations on F22n with coefficients inF2n .
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 forn ≥ 4.
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
The existence of APN permutations on F22n is a long-termopen problem.
Hou proved that there are no APN permutations over F24 andthere are no APN permutations on F22n with coefficients inF2n .
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 forn ≥ 4.
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
The existence of APN permutations on F22n is a long-termopen problem.
Hou proved that there are no APN permutations over F24 andthere are no APN permutations on F22n with coefficients inF2n .
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 forn ≥ 4.
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
The Kloosterman sum K (a) over F2n is defined for anya ∈ F2n by
K (a) =∑a∈F∗
2n
(−1)Tr(ax+1x )
Shin, Kumar and Helleseth found that the existence of certain3-designs in the Goethals code of length 2n, n odd, over Z4
was equivalent to the identity
K
(a
1 + a4
)= K
(a3
1 + a4
)∀a ∈ F2n \ {1}
and they proved this identity for all odd values of n.This relation was extended to the case n even by Helleseth andZinoviev.
Zulfukar Saygı Permutation Polynomials over F.F.
Motivation
The Kloosterman sum K (a) over F2n is defined for anya ∈ F2n by
K (a) =∑a∈F∗
2n
(−1)Tr(ax+1x )
Shin, Kumar and Helleseth found that the existence of certain3-designs in the Goethals code of length 2n, n odd, over Z4
was equivalent to the identity
K
(a
1 + a4
)= K
(a3
1 + a4
)∀a ∈ F2n \ {1}
and they proved this identity for all odd values of n.This relation was extended to the case n even by Helleseth andZinoviev.
Zulfukar Saygı Permutation Polynomials over F.F.
Special PPs
Helleseth and Zinoviev used the PPs(1
x2 + x + δ
)2l
+ x over F2n
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
f (x) =(xp
i − x + δ)s
+ L(x)
over the finite field Fq have been extensively studiedwhere , i , s ∈ Z+, δ ∈ Fq, char(Fq) = p and L(x) is alinearized polynomial in Fq[x ].
Zulfukar Saygı Permutation Polynomials over F.F.
Special PPs
Helleseth and Zinoviev used the PPs(1
x2 + x + δ
)2l
+ x over F2n
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
f (x) =(xp
i − x + δ)s
+ L(x)
over the finite field Fq have been extensively studiedwhere , i , s ∈ Z+, δ ∈ Fq, char(Fq) = p and L(x) is alinearized polynomial in Fq[x ].
Zulfukar Saygı Permutation Polynomials over F.F.
Some PPs of the form(xp
i − x + δ)s
+ L(x)
T. Helleseth, V. Zinoviev, New Kloosterman sums identitiesover F2m for all m, Finite Fields Appl. 9 (2003) 187-193.
J. Yuan, C. Ding, Four classes of permutation polynomials ofF2m , Finite Fields Appl. 13 (2007) 869-876.
J. Yuan, C. Ding, H. Wang, J. Pieprzyk, Permutationpolynomials of the form (xp − x + δ)s + L(x), Finite FieldsAppl. 14 (2008) 482-493.
X. Zeng, X. Zhu, L. Hu, Two new permutation polynomialswith the form (x2
k+ 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 ofpermutation polynomials over finite fields, Finite Fields Appl.22 (2013) 16-23.
Z. Tu, X. Zeng, C.Li, T. Helleseth, Permutation polynomialsof the form (xp
m − x + δ)s + L(x) over the finite field Fp2m ofodd characteristic, Finite Fields Appl. 34 (2015) 20-35.
Zulfukar Saygı Permutation Polynomials over F.F.
Known Explicit PPs
Let p be an odd prime1 For positive integers n and k and δ ∈ Fpn ,(
xpk − x + δ
) pn+12
+ xpk
+ x is a PP of Fpn .
2 For positive integer k and δ ∈ F33k with Tr33k/3k (δ) = 0,(x3
k − x + δ) 33k−1
2 +3k
+ x3k
+ x is a PP of F33k .
3 For positive integers n and k with n|4k and δ ∈ Fpn ,(xp
k − x + δ) pn−1
2 +p2k
± (xpk
+ x) is a PP of Fpn .
4 For a positive integer m and for any δ ∈ F32m ,(x3
m − x + δ)2·3m−1
+ x3m
+ x is a PP of F32m .5 For a positive integer m and δ ∈ F32m , if (Tr32m/3m(δ))2 + 1 = 0
or a square in F3m ,(x3
m − x + δ)3m+2
+ x is a PP of F32m .
Zulfukar Saygı 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) = (x3( t−1
2 )k
− x + δ)s + x and
g(x) = (x3( t−1
2 )k
− x + δ)s + x3( t−1
2 )k
+ xare permutation polynomials over F3n with s = 3n−1
t + 1 andTr(δ) = 0.
Zulfukar Saygı Permutation Polynomials over F.F.
New PPs
Theorem
Let n = 4k , where k is a positive integer and δ ∈ F∗3n .f (x) = (x3
2k − x + δ)s + x is a permutation polynomial over F3n
for the following cases:
Let k be a positive integer, w be the generator of F3n ,
s = 3(3n−15 ) + 1 and ` = 2
⌊3n/2−1
5
⌋+ 3n/4.
Then for δ = w ` (mod 2`) ∈ F3n and Tr(δ) = 0, f (x) is a PPover F3n .
Let k = 1 and s = (3n−15 ) + 1. For each δ ∈ F∗3n with
Tr(δ) = 0, then f (x) is a PP over F3n .
Let k = 1 and s = 2(3n−15 ) + 1. Then for each δ ∈ F∗3n f (x) is
a PP over F3n .For this case f (x) + x and f (x) + x3
kare also a permutation
polynomial over F3n .
Zulfukar Saygı Permutation Polynomials over F.F.
New PPs
Theorem
Let n = 4k , where k is a positive integer and δ ∈ F∗7n .Let w be the generator of F7n , s = i(7
n−15 ) + 1, where i ∈ {1, 2, 3}
and ` = 2⌊7n/2−1
5
⌋+ 7n/4.
Then for δ = w ` (mod 2`) ∈ F7n and Tr(δ) = 0,
f (x) = (x72k − x + δ)s + x is a permutation polynomial over F7n .
Zulfukar Saygı Permutation Polynomials over F.F.
THANKS ...
Zulfukar Saygı Permutation Polynomials over F.F.