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Known infinite families of quadratic APN polynomials over GF(2^n)
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Revision as of 14:09, 4 January 2019 by
Fady
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(Created page with "<table> <tr> <th><math>N^\circ</math></th> <th>Functions</th> <th>Conditions</th> <th>References</th> </tr> <tr> <td>C1-C2</td> <td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s...")
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N
∘
{\displaystyle N^{\circ }}
Functions
Conditions
References
C1-C2
x
2
s
+
1
+
u
2
k
−
1
x
2
i
k
+
2
m
k
+
s
{\displaystyle x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}}
n
=
p
k
,
gcd
(
k
,
3
)
=
gcd
(
s
,
3
k
)
=
1
,
p
∈
{
3
,
4
}
,
i
=
s
k
mod
p
,
m
=
p
−
i
,
n
≥
12
,
u
primitive in
F
2
n
∗
{\displaystyle n=pk,\gcd(k,3)=\gcd(s,3k)=1,p\in \{3,4\},i=sk\mod p,m=p-i,n\geq 12,u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}
[1]
C3
s
x
q
+
1
+
x
2
i
+
1
+
x
q
(
2
i
+
1
)
+
c
x
2
i
q
+
1
+
c
q
x
2
i
+
q
{\displaystyle sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+cx^{2^{i}q+1}+c^{q}x^{2^{i}+q}}
q
=
2
m
,
n
=
2
m
,
g
c
d
(
i
,
m
)
=
1
,
c
∈
F
2
n
,
s
∈
F
2
n
∖
F
q
,
X
2
i
+
1
+
c
X
2
i
+
c
q
X
+
1
has no solution
x
{\displaystyle q=2^{m},n=2m,gcd(i,m)=1,c\in \mathbb {F} _{2^{n}},s\in \mathbb {F} _{2^{n}}\setminus \mathbb {F} _{q},X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1{\text{ has no solution }}x}
s.t.
x
q
+
1
=
1
{\displaystyle x^{q+1}=1}
[2]
C4
x
3
+
a
−
1
T
r
n
(
a
3
x
9
)
{\displaystyle x^{3}+a^{-1}\mathrm {Tr} _{n}(a^{3}x^{9})}
a
≠
0
{\displaystyle a\neq 0}
[3]
C5
x
3
+
a
−
1
T
r
n
3
(
a
3
x
9
+
a
6
x
18
)
{\displaystyle x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{3}x^{9}+a^{6}x^{18})}
3
|
n
{\displaystyle 3|n}
,
a
≠
0
{\displaystyle a\neq 0}
[4]
C6
x
3
+
a
−
1
T
r
n
3
(
a
6
x
18
+
a
12
x
36
)
{\displaystyle x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{6}x^{18}+a^{12}x^{36})}
3
|
n
,
a
≠
0
{\displaystyle 3|n,a\neq 0}
[4]
C7-C9
u
x
2
s
+
1
+
u
2
k
x
2
−
k
+
2
k
+
s
+
v
x
2
−
k
+
1
+
w
u
2
k
+
1
,
x
2
s
+
2
k
+
s
{\displaystyle ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1},x^{2^{s}+2^{k+s}}}
n
=
3
k
,
gcd
(
k
,
3
)
=
gcd
(
s
,
3
k
)
=
1
,
v
,
w
∈
F
2
k
,
v
w
≠
1
,
3
|
(
k
+
s
)
u
primitive in
F
2
n
∗
{\displaystyle n=3k,\gcd(k,3)=\gcd(s,3k)=1,v,w\in \mathbb {F} _{2^{k}},vw\neq 1,3|(k+s)u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}
[5]
C10
(
x
+
x
2
m
)
2
k
+
1
+
u
′
(
u
x
+
u
2
m
x
2
m
)
(
2
k
+
1
)
2
i
+
u
(
x
+
x
2
m
)
(
u
x
+
u
2
m
x
2
m
)
{\displaystyle (x+x^{2m})^{2^{k}+1}+u'(ux+u^{2m}x^{2m})^{(2^{k}+1)2^{i}}+u(x+x^{2m})(ux+u^{2m}x^{2m})}
n
=
2
m
,
m
⩾
2
{\displaystyle n=2m,m\geqslant 2}
even,
gcd
(
k
,
m
)
=
1
,
{\displaystyle \gcd(k,m)=1,}
and
i
⩾
2
{\displaystyle i\geqslant 2}
even
u
primitive in
F
2
n
∗
,
u
′
∈
F
2
m
not a cube
{\displaystyle u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*},u'\in \mathbb {F} _{2^{m}}{\text{ not a cube }}}
[6]
C11
a
2
x
2
2
m
+
1
+
1
+
b
2
x
2
m
+
1
+
1
+
a
x
2
2
m
+
2
+
b
x
2
m
+
2
+
(
c
2
+
c
)
x
3
{\displaystyle a^{2}x^{2^{2m+1}+1}+b^{2}x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^{2}+c)x^{3}}
n
=
3
m
,
m
odd
L
(
x
)
=
a
x
2
2
m
+
b
x
2
m
+
c
x
satisfies the conditions in lemma 8 of
[
3
]
{\displaystyle n=3m,m\ {\text{odd}}\ L(x)=ax^{2^{2m}}+bx^{2m}+cx\ {\text{satisfies the conditions in lemma 8 of}}\ [3]}
[7]
↑
L. Budaghyan, C. Carlet, G. Leander,
Two Classes of Quadratic APN Binomials Inequivalent to Power Functions
, IEEE Trans. Inform. Theory 54(9), 2008, pp. 4218-4229
↑
L. Budaghyan and C. Carlet.
Classes of Quadratic APN Trinomials and Hexanomials and Related Structures
. {\em IEEE Trans. Inform. Theory}, vol. 54, no. 5, pp. 2354-2357, 2008.
↑
L. Budaghyan, C. Carlet and G.Leander,
Constructinig new APN functions from known ones, Finite Fields and their applications
, vol.15, issue 2, Apr. 2009, pp. 150-159.
↑
4.0
4.1
L. Budaghyan, C. Carlet and G.Leander,
On a Construction of quadratic APN functions, Proceedings of IEEE information Theory workshop
ITW'09, Oct. 2009, 374-378.
↑
Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011).
A few more quadratic APN functions. Cryptography and Communications
, 3(1), 43-53.
↑
Göloğlu, Faruk.
Almost perfect nonlinear trinomials and hexanomials.
Finite Fields and Their Applications 33 (2015): 258-282.
↑
Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R.
On Isotopic Construction of APN Functions.
SETA 2018
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s.t. 
, 
even, 
and 
even 
![{\displaystyle n=3m,m\ {\text{odd}}\ L(x)=ax^{2^{2m}}+bx^{2m}+cx\ {\text{satisfies the conditions in lemma 8 of}}\ [3]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9e1622a7a32d5d862e198cc1128882c158b589)
