Cryptography Reference
In-Depth Information
7 Construction of Equivalence Classes with Desired
Properties
We now discuss how to construct equivalence classes with desired properties.
Let
f
B
n
M
1
b
0
|
,where
E
is suitable chosen so that the
equivalence class that
f
belongs to has desired properties. Clearly, any
g
∈
and 1
f
=
{
i
∈
E
}
∼
f
can be represented by
M
k
+
i
j
1
g
=
{
b
0
|
i
∈
E
}
,
φ
(2
n
2
n
where 1
≤
j
≤
−
1)
/n
,
M
j
is a generator matrix and 0
≤
k
≤
−
2. If we
want
AI
(
f
)
>r
, then we should investigate the following matrices
⎛
⎞
M
i
1
l
1
j
b
1
M
i
1
l
2
j
M
i
1
l
t
j
b
1
···
b
1
⎝
⎠
M
i
2
l
1
j
b
1
M
i
2
l
2
M
i
2
l
t
j
b
1
···
b
1
j
,
(2)
···
···
···
···
M
i
s
l
1
j
b
1
M
i
s
l
2
M
i
s
l
t
j
b
1
···
b
1
j
r
and
t
=
i
=0
i
,
i
1
<i
2
<
l
1
<l
2
< ... < l
t
<
2
n
where 0
≤
−
1,
wt
(
l
j
)
≤
... < i
s
are all in
E
or not and
s
=
2
n−
1
if
i
1
, ..., i
s
∈
E
2
n−
1
−
1if
i
1
, ..., i
s
/
∈
E.
It is easily found that
(
f
)
>r
if and only if all these matrices are of rank
t
.Let
M
j
b
0
=(
b
ij
1
,b
ij
2
, ..., b
ijn
)
T
.Ifwewantdeg(
f
)
AI
≥
d
, then we should investigate
the following functions
n
(
x
k
+
b
ijk
+1)+
c
1
·
x
1
x
2
···
x
n
,
i∈E
k
=1
where
c
1
= 0 or 1. Clearly, deg(
f
)
≥
d
if and only if all these functions are of
=2
n−
1
or 2
n−
1
degrees at least
n
−
1. Let
|
E
|
−
1. Then
nl
(
f
)=2
n
−
max
h∈A
n
(
|
1
f
∩
0
h
|
+
|
0
f
∩
1
h
|
)
=2
n
c
2
=2
n
−
2max
h∈A
n
|
1
f
∩
0
h
|−
−
2max
h∈A
n
|
0
f
∩
1
h
|−
c
2
.
where
c
2
=0or
±
1. If we want
nl
(
f
)tobehigh,thenmax
h∈A
n
(
|
1
g
∩
0
h
|
) should
be low, where
g
∼
f
. Therefore, we should investigate the sets
M
j
b
0
|
{
i
∈
E
}∩
0
h
and the number of elements of these sets should be small, where 1
≤
j
≤
φ
(2
n
A
n
. The equivalence classes to
which
f
1
or
f
2
belongs are examples with optimum algebraic degree, optimum
algebraic immunity and a good nonlinearity. In a similar way we can construct
other equivalence classes with desired properties. In fact, it may be not easy
−
1)
/n
,
M
j
is a generator matrix and
h
∈
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