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R
i
L
i
L
i
[all]
R
i
[all]
E
A1,A2
P
32/80
>>> 11
A
i
4
E
A3,A4
P
32/80
>>> 17
A
i
3
R
i
[all] A
i
4
[all]
R
i
[all] A
i
4
[all]
G
i
[all]
G
A3,A4
G
i
P
-1
32/80
E
A5,A6
>>> 11
L
i
[all]
R
i
[all] A
i
4
[all]
G
i
[all]
L
i+1
R
i+1
Fig. 5.
Linear property of SPECTR-H64
4.1 Degree of Boolean Functions
Let
X
=(
x
1
,
,x
n
) denote the plaintext, then the degree of a Boolean function
f,
deg
(
f
), is defined as the degree of the highest degree of the algebraic normal
form:
···
,x
n
)=
a
0
⊕
1
≤i≤n
a
i
x
i
⊕
1
≤i≤j≤n
f
(
x
1
,
···
a
i,j
x
i
x
j
⊕···⊕
a
1
,
2
,...,n
x
1
x
2
···
x
n
,
where
a
i
∈
GF
(2). Then, the degree of a vectorial Boolean function
F
(
x
1
,
···
,x
n
)=(
f
1
,
···
,f
n
) defined as
deg
(
F
)
max deg
(
f
i
).
4.2 Higher Order Differential Attack
Let
L
r
denote an
r
-dimensional subspace of
GF
(2)
n
.
GF
(2)
n
,
Proposition 1.
[6] Let f be a Boolean function. Then for any
w
∈
x∈L
r
+1
f
(
x
⊕
w
)=0
if and only if
deg
(
f
)
≤
r
.
The proposition 1 is valid for any vectorial Boolean function
F
. That is,
deg
(
F
)
⇔
x∈L
r
+1
F
(
x
≤
r
⊕
w
) = 0 for any
w
∈
GF
(2)
n
. We call
L
r
+1
(
r
+ 1)-th order differential structure.
Because of
deg
(
G
) = 3, we consider the fourth order differential structure
GF
(2)
32
.
G
(
x
⊕
w
)=0
,
for any w
∈
x∈L
4
We convert the above equation into
