permutation polynomials over finite...
TRANSCRIPT
![Page 1: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/1.jpg)
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.
![Page 2: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/2.jpg)
Outline
Basic Definitions and Notations
Some Known Results
Motivation
Some New Results and Open Problems
Zulfukar Saygı Permutation Polynomials over F.F.
![Page 3: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/3.jpg)
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.
![Page 4: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/4.jpg)
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.
![Page 5: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/5.jpg)
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.
![Page 6: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/6.jpg)
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.
![Page 7: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/7.jpg)
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.
![Page 8: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/8.jpg)
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.
![Page 9: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/9.jpg)
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.
![Page 10: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/10.jpg)
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.
![Page 11: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/11.jpg)
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.
![Page 12: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/12.jpg)
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.
![Page 13: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/13.jpg)
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.
![Page 14: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/14.jpg)
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.
![Page 15: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/15.jpg)
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.
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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.
![Page 17: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/17.jpg)
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.
![Page 18: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/18.jpg)
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.
![Page 19: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/19.jpg)
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.
![Page 20: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/20.jpg)
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.
![Page 21: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/21.jpg)
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.
![Page 22: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/22.jpg)
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.
![Page 23: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/23.jpg)
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.
![Page 24: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/24.jpg)
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.
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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.
![Page 26: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/26.jpg)
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.
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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.
![Page 28: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/28.jpg)
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.
![Page 29: Permutation Polynomials over Finite Fieldsmcs.bilgem.tubitak.gov.tr/cryptodays/files/2015-sunumlar/...polynomials of degree 6 or 7 over nite elds of characteristic 2, Finite Fields](https://reader036.vdocuments.us/reader036/viewer/2022081410/60a4b4037aecf0490758927b/html5/thumbnails/29.jpg)
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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 .
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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 .
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THANKS ...
Zulfukar Saygı Permutation Polynomials over F.F.